table of contents topic 7: polynomialsteachers.dadeschools.net/sdaniel/polynomial notes 201… ·...
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Topic 7: Polynomials
Table of Contents1. Introduction to Polynomials
2. Adding & Subtracting Polynomials
3. Multiplying Polynomials
4. Special Products of Binomials
5. Factoring Polynomials
6. Factoring Polynomials, part 2
7. Factoring by Grouping
Introduction to Polynomials
Monomial: a number, a variable, or a product of numbers and variables with whole‐number exponents.
Degree of a monomial: is the sum of the exponents of the variables. A constant has degree 0.
Vocab
Polynomial: an expression of more than two algebraic terms.Example: 3x4 + 5x2 – 7x + 1
Degree of a polynomial is the degree of the term with the greatest power/exponent.
Example: The degree of 3x4 + 5x2 – 7x + 1 is 4.
VocabFind the degree of each polynomial.
A. 11x7 + 3x3
B.
Degree of a Polynomial
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Standard form of a polynomial: Polynomial written with the terms in order from greatest degree to least degree.
Leading Coefficient: When written in standard form, the coefficient of the first term is called the leading coefficient.
Example: 3x4 + 5x2 – 7x + 1 and 3 is the leading coefficient.
VocabWrite the polynomial in standard form. Then give the leading coefficient.
1. 6x – 7x5 + 4x2 + 9
2. 16 – 4x2 + x5 + 9x3
3. 18y5 – 3y8 + 14y
Let’s Practice…
Special Polynomial Names
Degree Name
0
1
2
Constant
Linear
Quadratic
3
4
5
6 or more 6th,7th,degree and so on
Cubic
Quartic
Quintic
NameTermsMonomial
Binomial
Trinomial
Polynomial4 or more
1
2
3
By Degree
By # of Terms
You can add or subtract monomials by adding or subtracting like terms.
4a3b2 + 3a2b3 – 2a3b2
Like terms
Not like terms
The variables have the same powers.
The variables have different powers.
Review: Like Terms
Identify the like terms in each polynomial.
A. 5x3 + y2 + 2 – 6y2 + 4x3
B. 3a3b2 + 3a2b3 + 2a3b2 – a3b2
Like terms: ______________________
Identify Like Terms
Like terms: _______________________
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Adding and Subtracting Polynomials
Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.
Adding and Subtracting Polynomials
Like terms are constants or terms with the same variable(s) raised to the same power(s).
Remember!
Combine like terms.
A. 12p3 + 11p2 + 8p3
B. 5x2 – 6 – 3x + 8
Simplifying Polynomials
c. 2s2 + 3s2 + s – 3s2 – 5s
Combine like terms.
d. 4z4 – 8 + -2z2 +16z4 + 2 + 5z3 – 7
Let’s Practice…
Polynomials can be added in either vertical or horizontal form.
In vertical form, align the like terms and add:
In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms.
(5x2 + 4x + 1) + (2x2 + 5x + 2)
= (5x2 + 2x2 + 1) + (4x + 5x) + (1 + 2)
= 7x2 + 9x + 3
5x2 + 4x + 1
+ 2x2 + 5x + 2
7x2 + 9x + 3
2 Methods: Adding Polynomials
Add.
A. (4m2 + 5) + (m2 – m + 6)
B. (10xy + x) + (–3xy + y)
Adding Polynomials
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Add.
Let’s Practice…
Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a).
To subtract polynomials, remember that subtracting is the same as adding the opposite (distributing the negative). To find the opposite of a polynomial, you must write the opposite of each term in the polynomial:
–(2x3 – 3x + 7)= –2x3 + 3x – 7
Subtracting PolynomialsSubtract.
(–10x2 – 3x + 7) – (x2 – 9)
Subtracting Polynomials
Subtract.
(x3 + 4y) – (2x3)
Subtracting Polynomials
(7m4 – 2m2) – (5m4 – 5m2 + 8)
Subtract.
(2x2 – 3x2 + 1) – (x2 + x + 1)
Let’s Practice…
(9q2 – 3q) – (q2 – 5)
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Multiplying Polynomials
F.O.I.L
Multiplying Polynomials
Each term in the first polynomial, must be multiplied by each term in the second polynomial.
Method 1: Distribute
First
Outer
Inner
Last
• Multiply!!!
“F.O.I.L.”
Method 2: Box
Multiply (3x + 4)(5x ‐ 3)
1. Draw a box.
2. Write a polynomial on the top and side of a box.
3. Multiply.
4. Combine like terms.
3x +4
5x
-3
Pick Your Method: (7p ‐ 2)(3p ‐ 4)
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
21p2 – 34p + 8
7p -2
3p
-4
21p2
-28p
-6p
+8
21p2
-28p-6p+8
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Let’s Practice…
1. (7x – 10)(3x + 8)
2. (2x – 3)(4x ‐ 8)
3. (5x ‐10)(2x +8)
Multiplying Terms with Exponents
• When FOILing, add the exponents and multiply coefficients.
• Add the little numbers and multiply the big numbers!!!
Example:
(3x2 + 10x)(5x3 – 7x2)
15x5 ‐ 21x4 + 50x4 – 70x3
15x5 + 29x4 – 70x3
Let’s Practice…
1. (7x2 – 10x)(3x3 + 8x2)
2. (2x4 – 3x2)(4x ‐ 8)
3. (5x3 + 2x2)(8x ‐ 7)
Multiplying Larger Polynomials
Each term in the 1st polynomial must be multiplied by each term in the 2nd.
Example:
(7x2 + 2x + 8)(4x3 – 9x2)
Method 2:
Multiply: (2x ‐ 5)(x2 ‐ 5x + 4)
Let’s Practice…
1. (5x2 + 7) (2x3 – 5x2 +9)
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Special Products of Binomials
Multiply:
1. (2x + 3)(2x + 3)
2. (3x – 4)(3x + 4)
3. (x + 5)(x + 5)
Multiply:
4. (6x – 1)(6x – 1)
5. (3x + 2y)2
Special Products of BinomialsName Algebraically Words
Positive Perfect Square (a + b)2
Negative Perfect Square(a ‐ b)2
Difference of 2 Squares(a + b)(a – b)
Let’s Practice….1. (x + 8)2
2. (4x + 6y)2
3. (‐x + 5)2
4. (7x ‐ 3)(7x + 3)
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Factoring Polynomials
Vocab: Factoring
Factoring is rewriting an expression as a product of factors.
It is the reverse of multiplying polynomials FOILing.
2x x cb To determine the factors, ask yourself…
last number?!?!
What two #’s add to the middle number ANDmultiply to the last number?!?!
Let’s Practice…
2 3 2x x get 2?
What adds (or subtracts) to get 3 and multiplies to
get 2?
2 7 10x x multiplies to get 10?
What adds (or subtracts) to get ‐7 andmultiplies to get 10?
2 7 44x x What adds (or subtracts) to get ‐7 and multiplies to get
‐44?
Let’s Practice…
Factor:
1. x2 + 5x + 6
2. x2 ‐7x + 10
3. x2 ‐11x +24
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2x cbx
Signs of Factors
b c Factors
+ + +,+
‐ ‐+,‐ (The factor w/ the greater absolute value is ‐)
+ ‐+,‐ (The factor w/ the greater absolute value is +)
‐ + ‐, ‐
Vocab: GCF
The greatest common factor (GCF) is a common factor of the terms in the expression.
Example:
29 9 18x x
24 8 12x x
Vocab: Prime
If a polynomials is “prime” it means there are no factors.
Find the prime polynomials below:
1. x2 + 7x + 9
2. x2 + 5x + 4
3. x2 + 9x + 10
Let’s Practice….
Factor.
1. y2 ‐10y +16
2. r2 ‐11r +24
3. n2 ‐15n +56
4. 2v2 + 10v ‐72
Let’s Practice…
1. x2 + 12x + 36
2. x2 ‐ 8x + 16
3. ‐2x2 +22x ‐36
4. 16‐ x2
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Factoring Polynomials,
Part 2
(2 1)(3 5)x x 26 10 3 5x x x 26 13 5x x
Expanded Form
When factoring problems where a ≠ 1, we first want to get the problem into expanded form before we try to factor.
Expanded Form
Creating Expanded Form
Step 1: Multiply a∙c
Step 2: To get to expanded form ask yourself “What multiplies to get a∙c, and add/subtracts to get to b.”
Example:
1. Expand: 2x2 +9x +7
2. Expand: 3x2 + 2x – 8
Method 1:
Step 3: Write your new factors in place of bx.
Step 4: Group the first two terms together and the last two terms together.
Step 5: Factor each group
Step 6: Factor again to get the complete factorization
Method 1: 6x2 + 13 x +5 1) Multiply a∙c (6∙5=30)
2) To get to expanded form ask yourself “What multiplies to get a∙c, and add/subtracts to get to b.” (10, 3)
3) Write your new factors in place of bx. (6x2+10x+3x+5)
4) Group the first two terms together and the last two terms together. [(6x2+10x)+(3x+5)]
5) Factor each group [2x(3x+5)+1(3x+5)]
6) Factor again to get the complete factorization [(3x+5)(2x+1)]
Method 2: 6x2 + 13 x +5
Step 3: Fill in box.
Step 4: Factor horizontally and vertically.
Step 5: Terms outside of box are the solution.
Original 1st Term Expanded Term 1
Expanded Term 2 Original Last Term
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Factor: 2x2 + 5x ‐12 Original 1st Term Expanded Term 1
Expanded Term 2 Original Last Term
Factor: 3x2 + 7x +2
Factor: 2x2 + 15x ‐8 Factor: 16x2 + 28x +10
Factoring by Grouping
Factoring by Grouping – Using the distributive property to factor polynomials with four or more terms.
– Terms can be put into groups and then factored‐‐‐‐ each group will have a “like” factor used in regrouping.
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A polynomial can be factored by grouping if all of the following conditions exist.
1. There are four or more terms.
2. Terms have common factors that can be grouped together, and
3. There are two common factors that are identical.
Symbols: ax + bx + ay + by = (ax + bx) + (ay + by)
= x(a + b) + y(a + b)
= (x + y)(a + b)
Group, factor
Regroup
Factoring by Grouping Factor each polynomial by grouping. Check your answer.
6h4 – 4h3 + 12h – 8
Factor by Grouping
Factor each polynomial by grouping.
5y4 – 15y3 + y2 – 3y
Factor by GroupingFactor each polynomial by grouping.
6b3 + 8b2 + 9b + 12
Let’s Practice…
Factor each polynomial by grouping.
4r3 + 24r + r2 + 6
Let’s Practice…2x3 – 12x2 + 18 – 3x
Factoring with Opposite Groups