### Alg - 1 - 8.5

8-5 Factoring Special Products
Warm Up
Determine whether the following are
perfect squares. If so, find the square
root.
1. 64
3. 45
5. y8
7. 9y7
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yes; 8
no
yes; y4
no
2. 36
yes; 6
4. x2
yes; x
yes; 2x3
6. 4x6
8. 49p10 yes;7p5
8-5 Factoring Special Products
Find the degree of each polynomial.
A. 11x7 + 3x3
7
D. x3y2 + x2y3 – x4 + 2
5
B.
4
C. 5x – 6
1
The degree of a polynomial is the degree
of the term with the greatest degree.
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8-5 Factoring Special Products
Learning Targets
Students will be able to: Factor perfectsquare trinomials and factor the
difference of two squares.
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8-5 Factoring Special Products
A trinomial is a perfect square if:
• The first and last terms are perfect squares.
• The middle term is two times one factor
from the first term and one factor from
the last term.
9x2
3x
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•
+
12x
+
4
3x 2(3x • 2) 2 • 2
8-5 Factoring Special Products
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8-5 Factoring Special Products
Determine whether each trinomial is a perfect
square. If so, factor. If not explain.
9x2 – 15x + 64
3x

3x
2(3x

8) 8  8
2(3x  8) ≠ –15x.
9x2 – 15x + 64 is not a perfect-square trinomial
because –15x ≠ 2(3x  8).
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8-5 Factoring Special Products
Determine whether each trinomial is a perfect
square. If so, factor. If not explain.
81x2 + 90x + 25
9x
●
9x
2(9x
81x2 + 90x + 25
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●
5) 5 ● 5
 9 x  5
The trinomial is a perfect
square. Factor.
2
8-5 Factoring Special Products
Determine whether each trinomial is a perfect
square. If so, factor. If not explain.
36x2 – 10x + 14
6x  6x
???
The trinomial is not a perfect-square
because 14 is not a perfect square.
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8-5 Factoring Special Products
Determine whether each trinomial is a perfect
square. If so, factor. If not explain.
x2 + 4x + 4
x

x
2(x

2) 2

2
The trinomial is a perfect
square. Factor.
x  4x  4   x  2
2
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2
8-5 Factoring Special Products
A rectangular piece of cloth must be cut to
make a tablecloth. The area needed is
(16x2 – 24x + 9) in2. The dimensions of
the cloth are of the form cx – d, where c
and d are whole numbers. Find an
expression for the perimeter of the cloth.
Find the perimeter when x = 11 inches.
4x  3
16x2 – 24x + 9
4 x  4 x 2  4 x  3
16x2 – 24x + 9
 4 x  3
P  x   4  4x  3
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33
2
4x  3
4x  3
P 11  4  4 11  3  164"
4x  3
8-5 Factoring Special Products
In Chapter 7 you learned that the difference of two
squares has the form a2 – b2. The difference of two
squares can be written as the product (a + b)(a – b).
You can use this pattern to factor some polynomials.
A polynomial is a difference of two squares if:
•There are two terms, one subtracted from the
other.
• Both terms are perfect squares.
4x2 – 9
2x
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
2x
3

3
8-5 Factoring Special Products
Recognize a difference of two squares: the
coefficients of variable terms are perfect squares,
powers on variable terms are even, and
constants are perfect squares.
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8-5 Factoring Special Products
Determine whether each binomial is a difference
of two squares. If so, factor. If not, explain.
3p2 – 9q4
???
3p2 is not a perfect square.
3p2 – 9q4 is not the difference of
two squares because 3p2 is not a
perfect square.
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8-5 Factoring Special Products
Determine whether each binomial is a difference
of two squares. If so, factor. If not, explain.
100x2 – 4y2
10x  10x
2y

2y
The polynomial is a difference
of two squares.
Write the polynomial as
(a + b)(a – b).
100x2 – 4y2 = (10x + 2y)(10x – 2y)
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8-5 Factoring Special Products
Determine whether each binomial is a difference
of two squares. If so, factor. If not, explain.
x4 – 25y6
x2

x2
5y3  5y3
The polynomial is a difference
of two squares.
Write the polynomial as
(a + b)(a – b).
x4 – 25y6 = (x2 + 5y3)(x2 – 5y3)
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8-5 Factoring Special Products
Determine whether each binomial is a difference
of two squares. If so, factor. If not, explain.
1 – 4x2
1

1
2x
The polynomial is a difference
of two squares.

2x
Write the polynomial as
(a + b)(a – b).
1 – 4x2 = (1 + 2x)(1 – 2x)
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8-5 Factoring Special Products
Determine whether each binomial is a difference
of two squares. If so, factor. If not, explain.
p8 – 49q6
p4
●
p4
7q3 ● 7q3
The polynomial is a difference
of two squares.
Write the polynomial as
(a + b)(a – b).
p8 – 49q6 = (p4 + 7q3)(p4 – 7q3)
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8-5 Factoring Special Products
Check It Out! Example 3c
Determine whether each binomial is a difference
of two squares. If so, factor. If not, explain.
16x2 – 4y5
16x2 – 4y5
4x  4x
4y5 is not a perfect square.
16x2 – 4y5 is not the difference of two squares
because 4y5 is not a perfect square.
HW pp. 562-564/13-29 odd,46-50,55-64
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