ee004 fhs lnt 003 chapter 1 jan11
TRANSCRIPT
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Learning Outcome:
1. Understand the definition of polynomials2. Add and subtract polynomials
3. Multiply polynomial
4. Use special products rule in polynomial multiplication
5. Use long division to divide polynomial
6. Solve the quadratic equations with FactorizationMethod, Completing the Square and Quadratic Formula
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Definition A polynomial is a single term or the sum of two or more
terms containing variables with whole number exponents.
A polynomial in the variablex is an algebraic expressionof the form:
anxn + an-1x
n-1+a1x + a0x0
where an 0; n is a nonnegative integer and coefficients a0,a1, a2 an are constant polynomial.x represents a real
number. Rules: Variables should have only nonnegativeand non-fraction integralexponentsand no variable isdenominator
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Classifying Polynomials
Monomiala polynomial with exactly one term
Binomiala polynomial that has two terms
Trinomiala polynomial with three terms
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Example 1 is a polynomial.
is not a polynomial because the second term
involves the division by the variable ofx
. is not a polynomial because the third
term contains an exponent which is not an non-negative integer.
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Exercise 1Determine whether the following statement is a
polynomial or not.
I. x2 +x1/2 + 9
II. x34x2 +
III. x-2 + 6x + 4
IV. 4x4
+ 2x + 7V. 20x3
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Standard form and degree A polynomials is in standard form if its terms are
arranged so that the powers of the variables are indescending orascending order
anxn + an-1x
n-1+a1x + a0a0 + a1x+ an-1x
n-1 + anxn
Degree of a polynomial equation is referred to the largestpower of the variable.
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Exercise 2Polynomial Degree Standard Form
x3 - 3x + 12
x10 -x208x
32 +x6
7x4
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Adding and Subtracting Polynomials are added and subtracted by combining like
terms
(ax + b) + (cx + d) = (a + c)x + (b + d) Following is the steps of solving the addition and
subtraction of polynomials
1. Rearrange like terms in the polynomials.
2. Add or subtract the like terms.
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Example 2
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Example 3
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Exercise 3 Solve the following problems, then write the
resulting polynomial in standard form.
1. (-6x3 + 5x2 - 8x + 9) + (17x3 + 2x2 - 4x -13)
2. (5x2 + 7x - 9) - (3x3 - 8) - (-x2 - 6x + 3)
3. (5x4 - 5x) + (8x - 7) - (2x2 - 3x - 9)
4. (6x4 + 2x) - (3x + 3x4)5. (x3 - 2x2 + 3x + 4) + (5 + 6x - 7x2 - 8x3)
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Multiplication of Polynomial In multiplication of polynomials, use the distributive
property to multiply each term by term, combine the liketerms and write the result in simplest form.
E.g:
ax(bx + c) = ax(bx) + ax(c)
If the multiplication operation involves the exponentfunctions, we have to add the powers of the polynomialfor multiplication.
E.g:
(ax2)bx5 = (ab)x2+5 = abx7
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Multiplication of Polynomial Use the distributive property to multiply the trinomial
by each term of the binomial. Multiply the monomials,then combine like terms.
(ax2 - bx + c)(dx- e)
= dx(ax2 - bx + c) -e(ax2 - bx + c)
= adx3 - bdx2 + dcx - aex2 + bex - ce
= adx3 + (- bd - ae)x2 + (dc+be)x - ce
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Example 4 Find the product of the following problem
Solution:
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Special Product Rules Certain types of products are important to deserve special
attention. These are stated in the following list of specialproduct rules. Each of them can be verified by directmultiplication.
Square of Binomial:(a + b)2 = a2 + 2ab + b2 Binomial Sum(a - b)2 = a22ab +b2 Binomial Difference
Sum and Difference of Two Squares:
(a - b)(a + b) = a2
b2
Cube of Binomial:(a + b)3 = a3 + 3a2b + 3ab2 + b3(a - b)3 = a33a2b + 3ab2b3
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Exercise 4
Find each product
1. (x + 4y)(3x - 5y)
2. (5x + 3y)2
3. (x + 1)(x2 -x +1)
4. (3x + 4)3
5. (2x + 4)(5x2 - 1)
6. (x - 1)(x - 3)
7. (2x)(x
2
+ 2x - 5)8. (6x + 5)(7x - 2)2
9. (5x - 3)3
10. (9x + 2y)(9x - 2y)
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Division of PolynomialsPolynomials division is one of the basic tools needed in the
study of the theory of polynomials. This subtopic willreviews on long division polynomials.
Long Division of PolynomialsIn order to perform the long division operation, the degreeof polynomials should be checked to be arranged indescending order.
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Example 5
Divide byand then write in the forms
Divisor
Remainderquotient
Divisor
Dividend
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Solution4y2 - 3y + 2 ) 8y3 - 18y2 + 11y - 6
Divisor
Dividend
- (8y3 - 6y2 + 4y)
-12y2 + 7y - 6-(-12y2 + 9y - 6)
-2y
2y Quotient
Remainder
234
232
234
61118822
23
yy
yy
yy
yyySo,
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Exercise 5Find the Quotient and Remainder of the following:
1. (x5 + 4x2 + 2x4 - 5x - 3x3 + 2) (x1 )
2. (2x3
3x2
+ 4x5)
(x2)3. (3x3x + 84x2 ) (4 + x )
4. (x5 + 10x2 + 4x37x4x5 ) (x1)
5. (4x339x28x ) (x3)
6. (x34x + 82x2 ) (2 + x)
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Quadratic Polynomial An expression of the form , where a, b and c
are constants with , is called a quadratic expression.
An equation of the form , where a, b and c
are constants with , is called a quadratic equation.
Quadratic equations can be solved by three methods,namely,
a) Factorization
b) Completing the square
c) Quadratic formula
cbxax 2
0a
02 cbxax0a
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Factorization
Example:
Solve the quadratic equations below by using factorization
method.
Solution:
x =0 ; 2x + 5 = 0
x =
baxxbxax 2
0522
xx
052
0522
xx
xx
2
5
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FactorizationIf
Note:
Ifc is positive, thenp and q must both have the same sign. Ifc is negative, thenp and q have opposite signs.
We know that a = mn, b = np + mq and c =pq
a) Ifa> 0 (positive) and c > 0 (positive),
i) when b > 0 (positive), thenp and q are positive,
ii) when b < 0 (negative), thenp and q are negative.
qnxpmxcbxax 2
pqxnpmqmnx
pqnpxmqxmnx
2
2
nx
mx p
pq
nx p
(np + mq)x
q
2mnx
mx q
Continue to next slide
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Factorizationb) Ifa> 0 (positive) and b and c have opposite signs, thenp and
q are also of opposite signs because m and n are positive.
If , then
i) ifb is positive, q is positive,ii) ifb is negative, q is negative.
If , then
i) ifb is positive, q is negative,
ii) ifb is negative, q is positive.
Continue to next slide
pqxnpmqmnxcbxax 22
npmq
mqnp
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Example 6
Solve the quadratic equations below by using factorization
method.
a) b)
c)
Solution:a)
3x + 2 = 0 or x2 =0
x = x = 2
0443 2 xx 0101962 xx
0443 2 xx
0223 xxx
3x +2
-4
+2x
-4x
-2
23x
-6x
3
2
0164 2 x
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Example 7b)
-3x + 2 = 0 or 2x5 =0
x = x =
010196
2
xx 05223 xx
3
2
2x
-3x +2
-10
+4x
19x
-5
2
6x
+15x
2
5
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Example 8c)
2x+ 4 = 0 or 2x - 4 =0
x = x =
Alternate method:
0164
2
x 04242 xx
2
2x
2x +4
-10
+8x
0
-4
2
4x
-8x
2
0164 2 x
2
4
4
164
2
2
x
x
x
x
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NoteIfu is an algebraic expression and dis a positive real
number, u2 = dhas exactly two solutions
Ifu2 = d, then u = oru = -
Equivalently,
Ifu2 = d, then u =
E.g. 1:
5x2
= 20x2 =
x =
x =2
d
E.g. 2:
(x - 2)
2
6 = 0(x - 2)2 = 6
x2 =
x = 2
d
d
45
20
6
6
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Converting a quadratic polynomial tothe form of .
Ifx2 + bx is a binomial, then by adding , which is the
square of half the coefficient ofx, a perfect square
trinomial will result.That is,
Note:
If , then
Completing the Square
cbxaxxp 2
)(qpxaxp 2)()( 2
2
b
bxx222
22
bx
b
du 2 du
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Example 9 Solve by using completing the square.
Solution:
0362 xx
362 xx
0362 xx
222 3336 xx
123 2 x
123 x
323x
22
2
2
632
66
xx
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Example 10 Solve by using completing the square.
Solution:
0323 2 xx
03
3
3
23
2
xx
0323 2 xx
01
3
22 xx
13
22 xx
22
2
23
2
123
2
32
xx
22
2
311
31
32
xx
Continue at next slide
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Example 10 (cont.)
9
10
3
12
x
9
10
3
1x
3
101
3
10
3
1
9
10
3
1
x
x
x
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Quadratic formula
For any quadratic equations , the roots aregiven by
02
cbxax
a
acbbx
2
42
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Example 11 Solve by using quadratic formula.
Solution:
From the equation, we know,
a = 1, b = -7 and c = 12
Then,
01272 xx
12
1214772
x
2
48497 x
2
17x
4
2
17
x
x or
3
2
17
x
x
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Example 12 Solve by using quadratic formula.
Solution:
From the equation, we know,
a = 4, b = 12 and c = 7
Then,
07124 2 xx
42
74412122
x
8
11214412 x
8
3212 x
2
23
8
2412
x
x or
8
21612 x
8
21612 x
8
2412 x
2
23
8
2412
x
x
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Exercise 61. Solve the following equations
a. b.
2. Solve the equations by using the factorization method.
a)
b)c)
d)
3. Solve the equations by completing the square, giving the
solutions in surd form.a)
b)
c)
022 xx
0253
2
xx 092 xx9412 2 xx
0635 2 xx0652 xx
012 2 xx
092 2 x 0473 2 x
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3. Solve the equations by using quadratic formula.
a)
b)
c)
d)
Exercise 6 (cont.)
0169 2 xx
0842 xx
0962 2 xx
0425 2 xx