2.3 add, subtract, & multiply polynomials
DESCRIPTION
2.3 Add, Subtract, & Multiply Polynomials. p. 104 What are the two ways that you can add, subtract or multiply polynomials? Name three special product patterns. To add or subtract, add or subtract the coefficients of like terms! Vertical format :. Add 3x 3 +2x 2 -x-7 and x 3 -10x 2 +8. - PowerPoint PPT PresentationTRANSCRIPT
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2.3 Add, Subtract, & Multiply Polynomials
p. 104
What are the two ways that you can add, subtract or multiply polynomials?
Name three special product patterns.
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To add or subtract, add or subtract the coefficients of like terms!Vertical format :
Add 3x3+2x2-x-7 and x3-10x2+8.
3x3 + 2x2 – x – 7 + x3 – 10x2 + 8 Line up
like terms
4x3 – 8x2 – x + 1
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Horizontal format: Combine like terms
(8x3 – 3x2 – 2x + 9) – (2x3 + 6x2 – x + 1)=
(8x3 – 2x3)+(-3x2 – 6x2)+(-2x + x) + (9 – 1)=
6x3 + -9x2 + -x + 8 =
6x3 – 9x2 – x + 8
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Subtract polynomials vertically
a. Subtract 3x3 + 2x2 – x + 7 from 8x3 – x2 – 5x + 1 in a vertical format.
SOLUTION
a. Align like terms, then add the opposite of the
subtracted polynomial.
8x3 – x2 – 5x + 1– (3x3 + 2x2 – x + 7)
8x3 – x2 – 5x + 1+ – 3x3 – 2x2 + x – 7
5x3 – 3x2 – 4x – 6
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Subtract polynomials horizontally
Write the opposite of the subtracted polynomial,
then add like terms.
(4z2 + 9z – 12) – (5z2 – z + 3) = 4z2 + 9z – 12 – 5z2 + z – 3
= 4z2 – 5z2 + 9z + z – 12 – 3
= – z2 + 10z – 15
b. Subtract 5z2 – z + 3 from 4z2 + 9z – 12 in a horizontal
format.
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Examples: Adding & Subtracting
(9x3 – 2x + 1) + (5x2 + 12x -4) =
9x3 + 5x2 – 2x + 12x + 1 – 4 =
9x3 + 5x2 + 10x – 3
(2x2 + 3x) – (3x2 + x – 4)=
2x2 + 3x – 3x2 – x + 4 =
2x2 - 3x2 + 3x – x + 4 =
-x2 + 2x + 4
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Multiplying Polynomials: Vertically
(-x2 + 2x + 4)(x – 3)=
-x2 + 2x + 4× x – 3
3x2 – 6x – 12 -x3 + 2x2 + 4x
-x3 + 5x2 – 2x – 12
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Multiplying Polynomials : Horizontally
(x – 3)(3x2 – 2x – 4)=
(x – 3)(3x2)
+ (x – 3)(-2x)
+ (x – 3)(-4) =
(3x3 – 9x2) + (-2x2 + 6x) + (-4x + 12) =
3x3 – 9x2 – 2x2 + 6x – 4x +12 =
3x3 – 11x2 + 2x + 12
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Multiplying 3 Binomials :
(x – 1)(x + 4)(x + 3) =
FOIL the first two:
(x2 – x +4x – 4)(x + 3) =
(x2 + 3x – 4)(x + 3) =
Then multiply the trinomial by the binomial
(x2 + 3x – 4)(x) + (x2 + 3x – 4)(3) =
(x3 + 3x2 – 4x) + (3x2 + 9x – 12) =
x3 + 6x2 + 5x - 12
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Some binomial products appear so much we need to recognize the patterns!
Sum & Difference (S&D):
(a + b)(a – b) = a2 – b2
Example: (x + 3)(x – 3) = x2 – 9
Square of Binomial:
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
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Last PatternLast Pattern
Cube of a Binomial(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a – b)3 = a3 - 3a2b + 3ab2 – b3
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Example: Example: (a + b)3 = a3 + 3a2b + 3ab2 + b3
(x + 5)3 =
a=? b = ?
a = x and b = 5
x3 + 3(x)2(5) + 3(x)(5)2 + (5)3 =
x3 + 15x2 + 75x + 125
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Find the Product
3. (x + 2)(3x2 – x – 5)
SOLUTION
3x2 – x – 5
x + 2
6x2 – 2x – 103x3 – x2 – 5x
3x3 + 5x2 – 7x – 10
Multiply 3x2 – x – 5 by 2 .
Multiply 3x2 – x – 5 by x .
Combine like terms.
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Multiply
4. (a – 5)(a + 2)(a + 6)
SOLUTION
(a – 5)(a + 2)(a + 6) = (a2 – 3a – 10)(a + 6)
= (a2 – 3a – 10)a + (a2 – 3a – 10)6
= (a3 – 3a2 – 10a + 6a2 – 18a – 60)
= (a3 + 3a2 – 28a – 60)
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Multiply
5. (xy – 4)3
SOLUTION
(xy – 4)3 = (xy)3 – 3(xy)2 + 3(xy)(4)2 – (4)3
= x3y3 – 12x2y2 + 48xy – 64
a=?
b=?
a = xy
b = 4
(a – b)3 = a3 - 3a2b + 3ab2 – b3
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Petroleum
Since 1980, the number W (in thousands) of United States
wells producing crude oil and the average daily oil output
per well O (in barrels) can be modeled by
W = – 0.575t2 + 10.9t + 548 and O = – 0.249t + 15.4
where t is the number of years since 1980. Write a model for the average total amount T of crude oil
produced per day. What was the average total amount of crude oil produced per day in 2000?
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What are the two ways that you can add, subtract or multiply polynomials?
Horizontally or vertically
Name three special product patterns.
Sum and difference, square of a binomial, and cube of a binomial (p.105).
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SOLUTION
To find a model for T, multiply the two given models.
– 0.575t2 + 10.9t + 548
– 0.249t + 15.4
– 8.855t2 + 167.86t + 8439.20.143175t3 – 2.7141t2 – 136.452t
0.143175t3 – 11.5691t2 + 31.408t + 8439.2
Total daily oil output can be modeled by T = 0.143t3 – 11.6t2 + 31.4t + 8440 where T is measured in thousands of barrels. By substituting t = 20 into the model, you can estimate that the average total amount of crude oil produced per day in 2000 was about 5570 thousand barrels, or 5,570,000 barrels.
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The models below give the average depth D (in feet) of new wells drilled and the average cost per foot C (in dollars) of drilling a new well. In both models, t represents the number of years since 1980. Write a model for the average total cost T of drilling a new well.
Industry
D = 109t + 4010 C = 0.542t2 – 7.16t + 79.4To find a model for T, multiply the two given models.
2173.68t2 + 28711.6t + 318394
59.078t3 – 780.44t2 – 8654.6t
59.078t3 + 1392.98t2 – 20057t + 318394
0.542t2 – 7.16t + 79.4
109t + 4010
Total daily oil output
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2.3 Assignmentp. 107, 3-45 every
third problem