c hapter 6 6-6 special products of binomials. o bjectives find special products of binomials
TRANSCRIPT
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CHAPTER 6 6-6 Special products of binomials
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OBJECTIVES
Find special products of binomials.
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SPECIAL PRODUCT OF BINOMIALS
Imagine a square with sides of length (a + b):
The area of this square is (a + b)(a + b) or (a + b)2. The area of this square can also be found by adding the areas of the smaller squares and the rectangles inside. The sum of the areas inside is a2 + ab + ab + b2.
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SPECIAL PRODUCT OF BINOMIALS
This means that (a + b)2 = a2+ 2ab + b2. You can use the FOIL method to verify this:
(a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2
F L
I= a2 + 2ab + b2
A trinomial of the form a2 + 2ab + b2 is called a perfect-square trinomial. A perfect-square trinomial is a trinomial that is the result of squaring a binomial.
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EXAMPLE 1: FINDING PRODUCTS IN THE FORM (A + B)2
Multiply.
Solution:
A. (x +3)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
(x + 3)2 = x2 + 2(x)(3) + 32Identify a and b: a = x and
b = 3.
= x2 + 6x + 9
B. (4s + 3t)2
(a + b)2 = a2 + 2ab + b2
(4s + 3t)2 = (4s)2 + 2(4s)(3t) + (3t)2
= 16s2 + 24st + 9t2
Identify a and b: a = 4s and b = 3t.
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CHECK IT OUT! EXAMPLE 1
Multiply.
Sol:
A. (x + 6)2
= x2 + 12x + 36
B. (5a + b)2
Sol= 25a2 + 10ab + b2
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SPECIAL PRODUCT OF BINOMIALS
You can use the FOIL method to find products in the form of (a – b)2.
(a – b)2 = (a – b)(a – b) = a2 – ab – ab + b2
F L
I= a2 – 2ab + b2
A trinomial of the form a2 – ab + b2 is also a perfect-square trinomial because it is the result of squaring the binomial (a – b).
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EXAMPLE 2: FINDING PRODUCTS IN THE FORM (A – B)2
Multiply.
Sol: Identify a and b: a = x and b = 6.
(a – b)2 = a2 – 2ab + b2
Identify a and b: a = 4m and b = 10.4m – 10)2 = (4m)2 – 2(4m)(10) + (10)2
A. (x – 6)2
(a – b)2 = a2 – 2ab + b2
(x – 6)2 = x2 – 2x(6) + (6)2
= x2 – 12x + 36
B. (4m – 10)2
= 16m2 – 80m + 100
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CHECK IT OUT! EXAMPLE 2
Multiply a. (x – 7)2
Sol:
= x2 – 14x + 49 b. (3b – 2c)2
Sol:= 9b2 – 12bc + 4c2
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DIFFERENCE OF SQUARES
You can use an area model to see that (a + b)(a–b)= a2 – b2.
Begin with a square with area a2. Remove a square with area b2. The area of the new figure is a2 – b2.
Remove the rectangle on the bottom. Turn it and slide it up next to the top rectangle.
The new arrange- ment is a rectangle with length a + b and width a – b. Its area is (a + b)(a – b).
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DIFFERENCE OF SQUARES
So (a + b)(a – b) = a2 – b2. A binomial of the form a2 – b2 is called a difference of two squares.
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EXAMPLE 3: FINDING PRODUCTS IN THE FORM (A + B)(A – B)
Multiply. A. (x + 4)(x – 4) Sol: (a + b)(a – b) = a2 – b2Use the rule for (a +
b)(a – b). (x + 4)(x – 4) = x2 – 42 Identify a and b: a = x
and b = 4.
= x2 – 16
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EXAMPLE
B. (p2 + 8q)(p2 – 8q) Sol: (p2 + 8q)(p2 – 8q) = (p2)2 – (8q)2
= p4 – 64q2
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CHECK IT OUT !!
Multiply a. (x + 8)(x – 8) Sol: x2 – 64
b. (3 + 2y2)(3 – 2y2) Sol:= 9 – 4y4
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EXAMPLE 4: PROBLEM-SOLVING APPLICATION
Write a polynomial that represents the area of the yard around the pool shown below.
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SOLUTION
Area of yard = total area – area of pool
a = x2 + 10x + 25 (x2 – 4) –
= x2 + 10x + 25 – x2 + 4
= (x2 – x2) + 10x + ( 25 + 4)
= 10x + 29
The area of the yard around the pool is 10x + 29.
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CHECK IT OUT! EXAMPLE 4
Write an expression that represents the area of the swimming pool.
The area of the pool is 25.
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SUMMARY
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STUDENT GUIDED PRACTICE
Do problems 2,5,6,7,9,10,19 and 20 in your book page 437
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HOMEWORK
Do even problems from 21-38 in your book page 437