welcome to interactive chalkboard algebra 2 interactive chalkboard copyright © by the mcgraw-hill...
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Algebra 2 Interactive ChalkboardCopyright © by The McGraw-Hill Companies, Inc.
Send all inquiries to:
GLENCOE DIVISIONGlencoe/McGraw-Hill8787 Orion PlaceColumbus, Ohio 43240
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Lesson 5-1 Monomials
Lesson 5-2 Polynomials
Lesson 5-3 Dividing Polynomials
Lesson 5-4 Factoring Polynomials
Lesson 5-5 Roots of Real Numbers
Lesson 5-6 Radical Expressions
Lesson 5-7 Rational Exponents
Lesson 5-8 Radical Equations and Inequalities
Lesson 5-9 Complex Numbers
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Example 1 Simplify Expressions with Multiplication
Example 2 Simplify Expressions with Division
Example 3 Simplify Expressions with Powers
Example 4Simplify Expressions Using Several Properties
Example 5 Express Numbers in Scientific Notation
Example 6 Multiply Numbers in Scientific Notation
Example 7 Divide Numbers in Scientific Notation
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Multiplying Monomials:
• When multiplying monomials you must ADD exponents.
• Example: 2x3 3x5 2x x x 3x x x x x 6x8
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Commutative Property
Answer: Definition of exponents
Definition of exponents
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Answer:
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Try TheseMultiply the following monomials.
1. a2 • a6
3. (-3b3c)(7b2c2)
2. 3x2 • 7x4
4. 2x2(6y3)(2x2y)
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Try TheseMultiply the following monomials.
1. a2 • a6
3. (-3b3c)(4b2c2)
2. 3x2 • 7x4
4. 2x2(6y3)(2x2y)
a8 21x6
-12b5c3 24x4y4
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Dividing Monomials:
• When dividing monomials you must SUBTRACT exponents.
• Example:
•
• Cancel x’s 3xx 3x2
5
7
2
6
x
xxxxxx
xxxxxxx
2
6
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Subtract exponents.
Remember that a simplified expression cannot contain negative exponents.
Answer: Simplify.
1 1
1 1
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Answer:
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Try TheseDivide the following monomials.
1.
3.
2.
4.
5
62
an
na52
75
zy
zy
cba
cba73
335
9
324
523
30
)3(2
dc
dcdc
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Try TheseDivide the following monomials.
1.
3.
2.
4.
5
62
an
na52
75
zy
zy
cba
cba73
335
9
324
523
30
)3(2
dc
dcdc
an -y3z2
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Power to a Power:
• When raising a power to a power you must MULTIPLY exponents.
• Example: (x3)5 This means 5 groups of (x3). (x3) (x3) (x3) (x3) (x3)
• (xxx)(xxx)(xxx)(xxx)(xxx)
• x15
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Product to a Power:
• When raising a product to a power you raise every number/variable to that power.
• Example: (2x2y3) 6 (2x2y3) (2x2y3) (2x2y3) (2x2y3) (2x2y3) (2x2y3)
which can be written as:
(2xxyyy) (2xxyyy) (2xxyyy) (2xxyyy) (2xxyyy) (2xxyyy)
64x12y18
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Quotient to a Power
• When raising a quotient to a power you raise the numerator & denominator to that power.
12555
3
3
33xxx
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Power of a power
Answer:
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Power of a powerAnswer:
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Power of a product
Power of a quotient
Answer:
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Negative exponent
Power of a quotient
Answer:
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Simplify each expression.
a.
b.
c.
d.
Answer:
Answer:
Answer:
Answer:
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Try TheseSimplify each monomial.
1. (n4)4
3. (-2r2s) 3 (3rs2)
2. (2x)4
4. 3
34
42
)3
6(
yx
yx
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Try TheseSimplify each monomial.
1. (n4)4
3. (-2r2s) 3 (3rs2)
2. (2x)4
4. 3
34
42
)3
6(
yx
yx
n1616x4
-24r7n5
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Negative Exponents
• To make a negative exponent positive, move the number/variable that is being raised to that exponent from the numerator to the denominator or vice versa.
• Example: x-3 3
1
x
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Try TheseSimplify each monomial.
1.
3. (a3b3)(ab)-2
2.
4. 22)(
y
xy
62
42
5
15
yx
yx
62
42
5
15
yx
yx
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Try TheseSimplify each monomial.
1.
3. (a3b3)(ab)-2
2.
4. 22)(
y
xy
2
4
6
28x
x
62
42
5
15
yx
yx
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Method 1 Raise the numerator and the denominator to the fifth power before simplifying.
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Answer:
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Method 2 Simplify the fraction before raising to the fifth power.
Answer:
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Answer:
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Express 4,560,000 in scientific notation.
4,560,000
Answer: Write 1,000,000 as a power of 10.
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Express 0.000092 in scientific notation.
Use a negative exponent.Answer:
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Express each number in scientific notation.
a. 52,000
b. 0.00012
Answer:
Answer:
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Express the result in scientific notation.
Associative and Commutative Properties
Answer:
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Associative and Commutative Properties
Express the result in scientific notation.
Answer:
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Evaluate. Express the result in scientific notation.
a.
b.
Answer:
Answer:
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Divide the number of red blood cells in the sample by the number of red blood cells in 1 milliliter of blood.
Answer: There are about 1.66 milliliters of blood in the sample.
Biology There are about red blood cells in one milliliter of blood. A certain blood sample contains red blood cells. About how many milliliters of blood are in the sample?
number of red blood cells in sample
number of red blood cells in 1 milliliter
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Answer:
Biology A petri dish startedwith germs in it. A half hour later, there are
How many times asgreat is the amount a half hour later?
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Assignment:
Page 226 #26, 32, 36, 40
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Example 1 Degree of a Polynomial
Example 2 Subtract and Simplify
Example 3 Multiply and Simplify
Example 4 Multiply Two Binomials
Example 5 Multiply Polynomials
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Polynomials
• Polynomial: The sum of terms such as 5x, 3x2, 4xy, 5
• Polynomial Terms have variable and whole number exponents. There are no square roots of exponents, no fractional powers, and no variables in the denominators.
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Polynomials
6x-2 Not a polynomial term
Has a negative exponent
1/x2 Not a polynomial term
Has variable in the denominator.
Not a polynomial term
Has variable in radical.
4x2 Is a polynomial term
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Determine whether is a polynomial. If it is a polynomial, state the degree of the polynomial.
Answer: This expression is not a polynomial
because is not a monomial.
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Answer: This expression is a polynomial because each term is a monomial. The degree of the first term is 5 and the degree of the second term is 2 + 7 or 9. The degree of the polynomial is 9.
Determine whether is a polynomial.
If it is a polynomial, state the degree of the polynomial.
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Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial.
a.
b.
Answer: yes, 5
Answer: no
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Adding Polynomials
• Add: (2x2 - 4) + (x2 + 3x - 3)
1. Remove parentheses.
2. Identify like terms.
3. Add the like terms.
• (2x2 - 4) + (x2 + 3x - 3)
• = 2x2 - 4 + x2 + 3x - 3
• = 3x2 + 3x - 7
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Subtracting Polynomials
• Subtract: (2x2 - 4) - (x2 + 3x - 3) 1. Remove parentheses. 2. Change the signs of ALL of the terms being
subtracted. 3. Change the subtraction sign to addition. 4. Follow the rules for adding signed numbers.
• (2x2 - 4) - (x2 + 3x - 3)• (Change the signs of terms being subtracted)• = (2x2 - 4) + (-x2 - 3x + 3) • = 2x2 - 4 + -x2 - 3x + 3• = x2 - 3x - 1
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Simplify
Distribute the –1.
Group like terms.
Combine like terms.Answer:
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Simplify
Answer:
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Try TheseAdd or subtract as indicated.
1. (3x2 – x – 2) + (x2 + 4x – 9)
2. (5y + 3y2) + (– 8y – 6y2)
3. (9r2 + 6r + 16) – (8r2 + 7r + 10)
4. (10x2 – 3xy + 4y2) – (3x2 + 5xy)
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Try TheseAdd or subtract as indicated.
1. (3x2 – x – 2) + (x2 + 4x – 9)
2. (5y + 3y2) + (– 8y – 6y2)
3. (9r2 + 6r + 16) – (8r2 + 7r + 10)
4. (10x2 – 3xy + 4y2) – (3x2 + 5xy)
4x2 + 3x - 11
-3y – 3y2
r2 – r + 6
7x2 -8xy + 4y2
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Multiplying Polynomials
• Simply multiply each term from the first polynomial by each term of the second polynomial.
• Example:
• (x + 3)(x² + 2x + 4)
• = x³ + 2x² + 4x + 3x² + 6x + 12
• = x³ + 2x² + 3x² + 4x + 6x + 12
• = x³ + 5x² + 10x + 12
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Distributive Property
Answer:Multiply the monomials.
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Answer:
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Outer terms Inner terms Last termsFirst terms
Answer: Multiply monomials and add like terms.
+ + +
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Answer:
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Distributive Property
Distributive Property
Multiply monomials.
Answer: Combine like terms.
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Answer:
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Try TheseMultiply the polynomials.
1. 4b(cb – zd)
2. 2xy(3xy3 – 4xy + 2y4)
3. (3x + 8)(2x + 6)
4. (x – 3y)2
5. (x2 + xy + y2)(x – y)
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Try TheseMultiply the polynomials.
1. 4b(cb – zd)
2. 2xy(3xy3 – 4xy + 2y4)
3. (3x + 8)(2x + 6)
4. (x – 3y)2
5. (x2 + xy + y2)(x – y)
4b2c – 4bdz
6x2y4 – 8x2y2 + 4xy5
6x2 + 34x + 48
x2 – 6xy + 9y2
x3 – y3
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Assignment:
Page 231-232 #25, 26, 30, 42
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Example 1 GCF
Example 2 Grouping
Example 3 Two or Three Terms
Example 4 Quotient of Two Trinomials
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Factoring Lesson #1
• Greatest Common Factor
• Polynomials in the form x2 + bx + c
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Greatest Common Factor
• The first thing you should always do when factoring is to take out a common factor. This is the simplest technique of factoring, but it is important even when you learn fancier techniques, because you will make your later work much easier if you always look for common factors first. Taking out common factors is using the distributive property backwards. The distributive property says:
a(b+c)=ab+ac
• The idea behind taking out a common factor is to look for something that all terms have “in common.” Look at thr right side of the above equation. There is a common factor, a.
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Greatest Common Factor
• A good trick for finding the greatest common factor to factor polynomials is to find the greatest common factor of the numbers and the smaller power of the variable, so here the greatest common factor of the numbers is 4 and the smallest power of x is 3, so we can take out 4x3 as a common factor.
Example:The polynomial:
4x5+12x4-8x3
Can be factored into:4x3(x2+3x-8)
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Example 1:
Factor the polynomial:
2x2 + 6x4
by taking out a common factor.
Solution: Choose the common factor. 2x2.
2x2 (1 + ___ )
2x2 (1 + 3x2)
Now Check your work:
2x2 (1 + 3x2)
Multiply back together:
2x2 + 6x4
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Example 2:
Factor the polynomial:
15x2y – 10xy2
by taking out a common factor.
Solution: Choose the common factor: 5xy.
5xy (3x – ___ )
5xy (3x – 2y)
Now Check your work:
5xy (3x – 2y)
Multiply back together:
15x2y – 10xy2
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Example 3:
Factor the polynomial:16a3b 5 – 24a2b4 – 8a4b7c
by taking out a common factor.
Solution: Choose the common factor: 8a2b4.
8a2b4 (2ab – ___ – ___ ) 8a2b4 (2ab – 3 – ___ )
8a2b4 (2ab – 3 – a2b3c )
Now Check your work: 8a2b4 (2ab – 3 – a2b3c )
Multiply back together:
16a3b 5 – 24a2b4 – 8a4b7c
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Now Try These:
Factor the following polynomials and check your work.
a. 6x2y3 + 8x2y5 Solution: 2x2y3 (3 + 4y2)
b. 12a4b2c3 – 18ab2c4 + 24a5b3c4
Solution: 6ab2c3 (2a3 – 3c + 4a4bc)
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Factoring Polynomials in the form x2 + bx + c (General Quadratics)
Examples of these “General Quadratics” are:
a. x2 + 7x + 10 b. x2 + 13x - 30
c. x2 - 8x + 15 d. x2 - 8x - 20
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Rules for Factoring General Quadratics
If the constant term is positive:
- Choose factors of the constant term whose SUM is the middle term.
- Use the same signs – the sign of the middle term.
• Example:
x2 + 10x + 16
( x )( x ) Choose factors of 16 whose sum is 10
(8 and 2)
( x 8 )( x 2 )
Use the same signs – sign of middle term (+)
( x + 8 )( x + 2 )
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Rules for Factoring General Quadratics
If the constant term is negative:
- Choose factors of the constant term whose DIFFERENCE is the middle term.
- Use different signs – the larger factor gets the sign of the middle term.
• Example:
x2 - 2x - 24
( x )( x ) Choose factors of 24 whose difference
is 2 (6 and 4)
( x 6 )( x 4 )
Use different signs – the six gets the sign of middle term (-)
( x - 6 )( x + 4 )
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Now Try These:
Factor the following polynomials and check your work.
a. x2 + 7x + 10 b. x2 + 13x - 30 Answer: (x + 5)(x + 2) Answer: (x + 15)(x – 2)
c. x2 - 8x + 15 d. x2 - 8x – 20
Answer: (x - 5)(x – 3) Answer: (x – 10)(x + 2)
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Part I
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Warm Up – Section 5-4 #1
Factor and check.
1. 10x2y + 15x3y2
2. 16a2b4c5 + 48a3bc2 – 12ab4c3
3. y2 + 11y + 24
4. y2 - 15y + 36
5. y2 + 7y - 30
6. y2 - 4y - 45
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Factoring Lesson #2
• Polynomials in the form ax2 + bx + c
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Factoring Polynomials in the form ax2 + bx + c (Trial and Error)
Examples of these “Trial and Error” Quadratics are:
a. 4x2 - 8x - 45 b. 12x2 + 13x - 14
c. 15x2 - 26x + 7 d. 25x2 +15x + 2
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Rules for Factoring General Quadratics in the form ax2 + bx + c
- List all of the possible factors of the first term and the last term.
- Choose the combination that will allow you to get the correct middle term.
- Check your work!!!
• Example: 4x2 - 24x + 35
4 x 1 5 x 72 x 2 35 x 1
Choose factors whose combination will give you the middle term (-24x). You may have to try different combinations before finding the one that works.
( 2x )( 2x )
Use the same signs – sign of middle term (-)
( 2x - 7 )( 2x - 5 )
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Now Try These:
Factor the following polynomials and check your work.
a. 2x2 + 7x + 6 b. 3x2 + 10x + 3 Answer: (2x + 3)(x + 2) Answer: (3x + 1)(x + 3)
c. 15x2 - 38x + 7 d. 10x2 - 3x – 27
Answer: (5x - 1)(3x – 7) Answer: (5x – 9)(2x + 3)
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Part II
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Warm Up – Section 5-4 #2
Factor using trial and error or the junk method and check your work.
1. 2x2 + 11x + 142. 14y2 – 19y – 3 3. 3a2 – 22a + 24
Factor.4. 25r2s4t + 100rs2t3
5. x2 – 11x + 24 6. x2 + 2x – 35
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Factoring Lesson #3
• Difference of two perfect squares x2 - y2
• Factoring by grouping
• Factoring Completely
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Factoring the difference of two perfect squares
Examples of these polynomials are:
a. 4x2 – 9 b. 16x2 – 36
c. x2 – 4 d. 25x2 – 16y2
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Rules for Factoring the difference of two perfect squares
- The square root of the first term becomes the first term of each binomial.
- The square root of the second term becomes the second term of each binomial.
- Use different signs.
Example: x2 - 64
Since the square root of x2 is x, x is the first term of each binomial.
( x )( x )
Since the square root of 64 is 8, 8 is the second term of each binomial.
( x 8 )( x 8 )
Use different signs.
( x + 8 )( x - 8 )
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Now Try These:
Factor the following polynomials and check your work.
a. 4x2 – 9 b. 16x2 – 36 Answer: (2x + 3)(2x - 3) Answer: (4x + 6)(4x - 6)
c. x2 – 4: d. 25x2 – 16y2
Answer: (x - 2)(x + 2) Answer: (5x – 4y)(5x + 4y)
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Factoring by grouping
Examples of polynomials that are factored by grouping are:
a. 6x2 + 3xy + 2xz + yz
b. 6x2 + 2xy – 3xz – yz
Note: You will see 4 terms when using the grouping method.
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Rules for Factoring by grouping:
- Group terms so that there is a common factor in each group.
- Take out the common factor in both groups.
- Combine like groups.
Example: 10a2 + 2ab + 5ad + bd
I will group the first two terms and the last two terms since both of those groups contain a common factor. Note: I am adding these groups.
(10a2 + 2ab) + (5ad + bd)
Take out a common factor.
2a(5a + b) + d(5a + b)
Combine like groups:
( 2a + d )( 5a + b )
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Examples of factoring by grouping:
Factor the polynomial:a3 – 4a2 + 3a – 12
Group:(a3 – 4a2)+ (3a – 12)Factor:a2 (a – 4) + 3(a – 4)Combine:(a2 + 3)(a – 4)
Factor the polynomial:7ac2 + 2bc2 – 7ad2 – 2bd2
Group:(7ac2 + 2bc2) + (– 7ad2 – 2bd2)Factor:c2(7a + 2b) + d2 (– 7a – 2b)Factor a negative out of second
group:c2(7a + 2b) - d2 ( 7a + 2b)Now groups match – so,
Combine:(c2 - d2) ( 7a + 2b)
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Now Try These:
Factor the following polynomials and check your work.
a. 6x2 + 3xy + 2xz + yz Answer: (3x + z)(2x + y)
b. 6x2 + 2xy – 3xz – yz
Answer: (2x - z)(3x + y)
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Factoring Completely
Some polynomials can be factored more than once. This may not be apparent from the beginning.
Just as integers can be factored into primes, polynomials can too, and it may take more than one step.
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Rules for Factoring Completely
- Factor a polynomial using the appropriate method.
- Check each factor to see if you can factor it again.
- If so, do it until all polynomials are prime.
Example: 3x2 – 21x + 30
Here, I notice that I have a common factor of 3, so take it out.
3(x2 – 7x + 10)
Now x2 – 7x + 10 can be factored.
3(x – 5)(x – 2)
Now all terms are prime.
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Now Try These:
Factor the following polynomials and check your work.
a. 2x2 + 12x + 18 b. 3x2 – 21x – 54 Answer: 2(x + 3)(x + 3) Answer: 3(x + 2)(x - 9)
c. 5x2 – 20: d. 25x2 – 100y2
Answer: 5(x - 2)(x + 2) Answer: 25(x – 2y)(x + 2y)
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Part III
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Warm Up Section 5-4 #3
Factor.
1. 7c3 – 28c2d + 35cd3
2. x2 – 5x – 14
3. x2 – 15x + 54
4. 3x2 – 22x + 35
5. 64x2 – 81
6. 3r + 3s + 5r3s + 5r2s2
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The GCF is 5ab.
Answer: Distributive Property
Factor
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Answer:
Factor
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Group to find the GCF.
Factor the GCF of each binomial.
Factor
Answer:Distributive Property
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Answer:
Factor
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Rewrite the expression using –5y and 3y in place of –2y and factor by grouping.
To find the coefficient of the y terms, you must find two numbers whose product is 3(–5) or –15 and whose sum is –2. The two coefficients must be 3 and –5 since
and .
Substitute –5y + 3y for –2y.
Factor
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Factor out the GCF of each group.
Answer:Distributive Property
Associative Property
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Answer: p2 – 9 is the difference of two squares.
Factor out the GCF.
Factor
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This is the sum of two cubes.
Answer:Simplify.
Sum of two cubes formula with and
Factor
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This polynomial could be considered the difference of two squares or the difference of two cubes. The difference of two squares should always be done before the difference of two cubes.
Difference of two squares
Answer:Sum and differenceof two cubes
Factor
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Factor each polynomial.a.
b.
c.
d.Answer:
Answer:
Answer:
Answer:
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Factor the numerator and the denominator.
Divide. Assume a –5, –2.
Answer: Therefore,
Simplify
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Simplify
Answer:
![Page 110: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/110.jpg)
![Page 111: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/111.jpg)
Example 1 Divide a Polynomial by a Monomial
Example 2 Division Algorithm
Example 3 Quotient with Remainder
Example 4 Synthetic Division
Example 5 Divisor with First Coefficient Other than 1
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Steps for Dividing a Polynomial by a Monomial
• 1. Divide each term of the polynomial by the monomial.
a) Divide numbersb) Subtract exponents
• 2. Remember to write the appropriate sign in between the terms.
Example:
Answer:
![Page 113: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/113.jpg)
Sum of quotients
Divide.
Answer:
![Page 114: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/114.jpg)
Answer:
![Page 115: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/115.jpg)
Try TheseDivide the following polynomials.
1.
2.
![Page 116: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/116.jpg)
Try TheseDivide the following polynomials.
1.
2.
332 32 yxxyx
332 532 abaab
![Page 117: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/117.jpg)
Use factoring to find
Answer:
![Page 118: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/118.jpg)
Answer: x + 2
Use factoring to find
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Try TheseDivide the following polynomials by factoring.
1. 2.
3. 4.
![Page 120: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/120.jpg)
Try TheseDivide the following polynomials by factoring.
1. 2.
3. 4.
2x 2x
3x 5x
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![Page 122: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/122.jpg)
Warm Up Section 5-5Covering lessons 5.1-5.4
1. Simplify: 5x2y(4x4y3)
2. Simplify: (4a5b4c2) 3
3. Simplify:
4. Multiply: (3x + 7)(x – 4)
5. Factor: x2 – 11x + 18
6. Factor: 3x2 + 7x – 20
84
36
12
18
yx
yx
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Warm Up Section 5-5Covering lessons 5.1-5.4
1. Simplify: 5x2y(4x4y3) 20x6y4
2. Simplify: (4a5b4c2) 3 64a15b12c6
3. Simplify:
4. Multiply: (3x + 7)(x – 4) 3x2 – 5x – 28
5. Factor: x2 – 11x + 18 (x – 9)(x – 2)
6. Factor: 3x2 + 7x – 20 (3x – 5)(x + 4)
84
36
12
18
yx
yx5
2
2
3
y
x
![Page 124: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/124.jpg)
Example 1 Find Roots
Example 2 Simplify Using Absolute Value
Example 3 Approximate a Square Root
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Simplifying Radicals
• When working with the simplification of radicals you must remember some basic information about perfect square numbers.
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Perfect Squares
1 = 1 x 1
4 = 2 x 2
9 = 3 x 3
16 = 4 x 4
25 = 5 x 5
36 = 6 x 6
49 = 7 x 7
64 = 8 x 8
81 = 9 x 9
100 = 10 x 10
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Perfect Squares Containing Variables
a2 = a x aa4 = a2 x a2
a6 = a3 x a3
a8 = a4 x a4
a10 = a5 x a5
So, a variable is a “perfect square” if it has an even exponent.
To take the square root, just divide the exponent by 2.
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Simplifying Radical Expressions
To simplify means to find another expression with the same value. It does not mean to find a decimal approximation.
Example: and, although it is equivalent to 5.65, we do not use the decimal value since the radical value is exact and the decimal is an estimate.
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To simplify (or reduce) a radical:• 1. Find the largest perfect
square which will divide evenly into the number under your radical sign. This means that when you divide, you get no remainders, no decimals, no fractions.
• 2. Write the number appearing under your radical as the product (multiplication) of the perfect square and your answer from dividing.
• 3. Give each number in the product its own radical sign.
• 4. Reduce the "perfect" radical which you have now created.
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Example:
• Reduce : the largest perfect square that divides evenly into 48 is 16.
• Find the largest perfect square which will divide evenly into 48.
• Give each number in the product its own radical sign.
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Example Continued
• Reduce the "perfect" radical which you have now created.
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Answer: The square roots of 16x6 are 4x3.
Simplify
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Simplify
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Answer: The fifth root is 3a2b3.
Simplify
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Answer: You cannot take the square root of a negative number.
Thus, is not a real number.
Simplify
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Answer: 3x4
Answer: 2xy2
Answer: not a real number
Answer:
Simplify.
a.
b.
c.
d.
![Page 137: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/137.jpg)
Try These
225 2)7( 3 27
16
1 25.0 4 8z
4)5( g 48169 yx 2)4( yx
![Page 138: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/138.jpg)
Try These
15 Not real # -3
1/4 0.5 z2
25g2 13x4y2 4x - y
225 2)7( 3 27
16
1 25.0 4 8z
4)5( g 48169 yx 2)4( yx
![Page 139: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/139.jpg)
Note that t is a sixth root of t6. The index is even, so the principal root is nonnegative. Since t could be negative, you must take the absolute value of t to identify the principal root.
Answer:
Simplify
![Page 140: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/140.jpg)
Since the index is odd, you do not need absolute value.
Answer:
Simplify
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Answer:
Simplify.
a.
b. Answer:
![Page 142: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/142.jpg)
Try These
169 2)4( 3 125
169
25 81.0 4 12z
8)2( x 10636 yx 2)63( x
![Page 143: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/143.jpg)
Try These
-13 4 5
5/13 0.9 z3
16x4 6x3y5 3x+6
169 2)4( 3 125
169
25 81.0 4 12z
8)2( x 10636 yx 2)63( x
![Page 144: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/144.jpg)
![Page 145: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/145.jpg)
Assignment:
• P248 #40, 42, 46, 50
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Example 1 Square Root of a Product
Example 2 Simplify Quotients
Example 3 Multiply Radicals
Example 4 Add and Subtract Radicals
Example 5 Multiply Radicals
Example 6Use a Conjugate to Rationalize a Denominator
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Factor into squares where possible.
Product Property of Radicals
Answer: Simplify.
Simplify
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Answer:
Simplify
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Simplify
Quotient Property
Factor into squares.
Product Property
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Rationalize the denominator.
Answer:
![Page 151: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/151.jpg)
Quotient Property
Rationalize the denominator.
Product Property
Simplify
![Page 152: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/152.jpg)
Multiply.
Answer:
![Page 153: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/153.jpg)
Answer:
Simplify each expression.
a.
b. Answer:
![Page 154: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/154.jpg)
Try These
72 3 54 4 96
3 316y 3 54242 nm
5 76
32
1zw
9
84
t
r10
520
y
x
5432 yx
![Page 155: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/155.jpg)
Try These
72 3 54 4 96
3 316y 3 54242 nm
5 76
32
1zw
9
84
t
r10
520
y
x
5432 yx
26 3 23 4 62
yyx 24 22 3 22y 3 234 mnmn
5 2
2
1wzwz 5
42
t
tr5
2 52
y
xx
![Page 156: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/156.jpg)
Warm Up 5-6Simplify.1.
2.
3.
4.
5.
6.
36
6281 yx
7681 yx
11440 yx
3 5324 ba
4 7432 ts
![Page 157: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/157.jpg)
Warm Up 5-6Simplify.1.
2.
3.
4.
5.
6.
36
6281 yx
7681 yx
11440 yx
3 5324 ba
4 7432 ts
639xy
yyx 339
yyx 102 52
3 232 bab
4 322 tst
![Page 158: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/158.jpg)
Factor into cubes.
Product Property of Radicals
Answer: Multiply.
Simplify
Product Property of Radicals
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Answer: 24a
Simplify
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Try These
1.
2.
)212)(123(
)205)(243(
![Page 161: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/161.jpg)
Try These
1.
2.
)212)(123(
)205)(243(
736
3060
![Page 162: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/162.jpg)
Product Property
Multiply.
Combine like radicals.
Simplify
Factor using squares.
Answer:
![Page 163: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/163.jpg)
Answer:
Simplify
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Try These
1.
2.
274812
54718024205
![Page 165: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/165.jpg)
Try These
1.
2.
274812
54718024205
33
62354
![Page 166: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/166.jpg)
F O I L
Product Property
Answer:
Simplify
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FOIL
Multiply.
Answer: Subtract.
Simplify
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Simplify each expression.
a.
b.
Answer: 41
Answer:
![Page 169: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/169.jpg)
Multiply by since
is the conjugate
of
FOIL
Difference ofsquares
Simplify
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Multiply.
Answer: Combine like terms.
![Page 171: Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION](https://reader035.vdocuments.net/reader035/viewer/2022062320/56649f4e5503460f94c6fd5a/html5/thumbnails/171.jpg)
Answer:
Simplify
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Assignment
• P 254 #16-46 even
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Example 1 Radical Form
Example 2 Exponential Form
Example 3Evaluate Expressions with Rational Exponents
Example 4Rational Exponent with Numerator Other Than 1
Example 5Simplify Expressions with Rational Exponents
Example 6 Simplify Radical Expressions
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Write in radical form.
Answer: Definition of
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Write in radical form.
Answer: Definition of
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Write each expression in radical form.
a.
b.
Answer:
Answer:
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Write using rational exponents.
Definition of Answer:
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Write using rational exponents.
Definition of Answer:
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Write each radical using rational exponents.
a.
b.
Answer:
Answer:
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Evaluate
Method 1
Answer: Simplify.
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Multiply exponents.
Method 2
Answer:
Power of a Power
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Answer: The root is 4.
Evaluate .
Method 1 Factor.
Power of a Power
Expand the square.
Find the fifth root.
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Answer: The root is 4.
Power of a Power
Multiply exponents.
Method 2
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Evaluate each expression.
a.
b.
Answer: 8
Answer:
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According to the formula, what is the maximum that U.S. Weightlifter Oscar Chaplin III can lift if he weighs 77 kilograms?
Answer: The formula predicts that he can lift at most 372 kg.
Weight Lifting The formula can be used to estimate the maximum total mass that a weight lifter of mass B kilograms can lift in two lifts, the snatch and the clean and jerk, combined.
Original formula
Use a calculator.
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Oscar Chaplin’s total in the 2000 Olympics was 355 kg. Compare this to the value predicted by the formula.
Answer: The formula prediction is somewhat higher than his actual total.
Weight Lifting The formula can be used to estimate the maximum total mass that a weight lifter of mass B kilograms can lift in two lifts, the snatch and the clean and jerk, combined.
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Answer: 380 kg
Answer: The formula prediction is slightly higher than hisactual total.
Weight Lifting Use the formula where M is the maximum total mass that a weight lifter of mass B kilograms can lift.
a. According to the formula, what is the maximum that a weight lifter can lift if he weighs 80 kilograms?
b. If he actually lifted 379 kg, compare this to the valuepredicted by the formula.
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Simplify .
Multiply powers.
Answer: Add exponents.
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Simplify .
Multiply by
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Answer:
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Simplify each expression.
a.
b.
Answer:
Answer:
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Simplify .
Rational exponents
Power of a Power
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Quotient of Powers
Answer: Simplify.
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Simplify .
Rational exponents
Power of a Power
Answer: Simplify.
Multiply.
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Answer: Multiply.
Simplify .
is the conjugate
of
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Answer: 1
Simplify each expression.
a.
b.
c. Answer:
Answer:
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Example 1 Solve a Radical Equation
Example 2 Extraneous Solution
Example 3 Cube Root Equation
Example 4 Radical Inequality
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Solving Radical Equations
1. Isolate the radical2. Raise each side to the
appropriate power to eliminate the radical.
3. Solve for the variable.
• Example:Solvefor x.
1. Isolate radical by adding 2 to both sides.
2.
Square both sides.
3.
So, x = 95
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Solve
Original equation
Add 1 to each side to isolate the radical.
Square each side to eliminate the radical.
Find the squares.
Add 2 to each side.
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Check
Answer: The solution checks. The solution is 38.
Replace y with 38.
Original equation
Simplify.
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Answer: 67
Solve
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Try TheseSolve each equation.
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Try TheseSolve each equation.
25 144
1 -11
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Solve
Original equation
Square each side.
Find the squares.
Isolate the radical.
Divide each side by –4.
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Answer: The solution does not check, so there is no real solution.
Check
Square each side.
Evaluate the squares.
Original equation
Evaluate the square roots.
Replace x with 16.
Simplify.
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Solve .
Answer: no real solution
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Solve
In order to remove the power, or cube root, you must
first isolate it and then raise each side of the equation to
the third power.
Original equation
Subtract 5 from each side.
Cube each side.
Evaluate the cubes.
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Answer: The solution is –42.
Divide each side by 3. Check
Original equation
Add.
Replace y with –42.
Simplify.
The cube root of –125 is –5.
Subtract 1 from each side.
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Answer: 13
Solve
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Try TheseSolve each equation.
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Try TheseSolve each equation.
49 5
9 -20
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Assignment
P 266 #16, 19, 22, 24
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Example 1 Square Roots of Negative Numbers
Example 2 Multiply Pure Imaginary Numbers
Example 3 Simplify a Power of i
Example 4 Equation with Imaginary Solutions
Example 5 Equate Complex Numbers
Example 6 Add and Subtract Complex Numbers
Example 7 Multiply Complex Numbers
Example 8 Divide Complex Numbers
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Keep in Mind:
• The square root of a negative number does not exist.
• Example: is not 5 or -5 since
5 x 5 = 25 and -5 x -5 = 25.
• So up until now, we could not simplify .
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i
i is defined to have the property that:
i2 = -1
therefore, we could say that square root of -1 is i.
This allows us to simplify the square roots of negative numbers such as .
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Examples
1. Simplify:
Since is 5 and is i, our answer is 5i.
2. Simplify
6ix2
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Simplify .
Answer:
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Simplify .
Answer:
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Simplify.
a.
b.
Answer:
Answer:
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Answer: = 6
Simplify .
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Answer:
Simplify .
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Answer: –15
Answer:
Simplify.
a.
b.
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Simplify
Multiplying powers
Power of a Power
Answer:
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Answer: i
Simplify .
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Solve
Answer:
Original equation
Subtract 20 from each side.
Divide each side by 5.
Take the square root of each side.
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Solve
Answer:
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Find the values of x and y that make the equationtrue.
Set the real parts equal to each other and the imaginary parts equal to each other.
Real parts
Divide each side by 2.
Imaginary parts
Answer:
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Find the values of x and y that make the equationtrue.
Answer:
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Simplify .
Answer:
Commutative and AssociativeProperties
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Simplify .
Commutative and Associative Properties
Answer:
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Simplify.
a.
b.
Answer:
Answer:
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Answer: The voltage is volts.
Electricity In an AC circuit, the voltage E, current I, and impedance Z are related by the formulaFind the voltage in a circuit with current 1 + 4 j ampsand impedance 3 – 6 j ohms.
Electricity formula
FOIL
Multiply.
Add.
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Electricity In an AC circuit, the voltage E, current I, and impedance Z are related by the formula E = I • Z. Find the voltage in a circuit with current 1 – 3 j ampsand impedance 3 + 2 j ohms.
Answer: 9 – 7 j
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andare conjugates.
Multiply.
Answer: Standard form
Simplify .
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Simplify .
Multiply.
Answer: Standard form
Multiply by
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Simplify.
a.
b.
Answer:
Answer:
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Explore online information about the information introduced in this chapter.
Click on the Connect button to launch your browser and go to the Algebra 2 Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to www.algebra2.com/extra_examples.
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End of Custom Shows
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custom shows and return to the main presentation.
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