Download Notes for Lesson 8-5: Factoring Special Products

Notes for Lesson 8-5: Factoring Special Products
8-5.1 – Recognizing and factoring perfect-square trinomials
A perfect square trinomial is a trinomial where the first and last terms are perfect squares and the middle term is
two times on factor from the first term and one factor from the last term.
PERFECT-SQUARE TRINOMIAL
a 2  2ab  b 2  (a  b)(a  b)  (a  b) 2
EXAMPLE
x  6 x  9  ( x  3)( x  3)  ( x  3) 2
a 2  2ab  b 2  (a  b)(a  b)  (a  b) 2
x 2  2 x  1  ( x  1)( x  1)  ( x  1) 2
2
Examples: Determine whether each trinomial is a perfect square. If so, factor.
x 2  12 x  36
4 x 2  12 x  9
x 2  9 x  16
ax
a  2x
b6
2ab  2( x)(6)  12 x
b3
2ab  2(2 x)(3)  12 x
( x  6) 2
(2 x  3) 2
ax
b4
2ab  1( x)(4)  8 x
Not a perfect  square
8-5.2 – Problem-Solving Applications
The park in the center of the Place des Vosges in Paris, France, is in the shape of a square. The area of the park
is 25 x 2  70 x  49 ft 2 . The side length of the park is in the form cx  d , where c and d are whole numbers.
Find the expression in terms of x for the perimeter of the park. Find the perimeter when x = 8 ft.


Area  Side 2
25 x 2  70 x  49
a  5x
b7
2ab  2(5 x)(7)  70 x
area  (5 x  7)
side  5 x  7
2
Perimeter  4( side)
Perimeter  4(5 x  7)
Perimeter  20 x  28
Perimeter if x  8 ft
Perimeter  20(8)  28
Perimeter  160  28
Perimeter  188 ft
8-5.3 – Recognizing and factoring the difference of two squares.
The difference of two squares occurs when you have a binomial where both terms are perfect squares and they
are being subtracted from each other.
DIFFERENCE OF TWO SQUARES
a 2  b 2  (a  b)(a  b)
EXAMPLE
x  9  ( x  3)( x  3)
2
Examples: Determine whether each binomial is a difference of two squares. If so, factor.
x 2  81
ax
b9
( x  9)( x  9)
9 p 4  16q 2
x6  7 y 2
a  3p2
b  4q
a  x3
(3 p 2  4q)(3 p 2  4q)
Do Practice B #’s 2, 4, 5, 7, 8, 9
b  7 y 2 is not a perfect square
Not the difference of two squares
1 4 x 2
Not subtraction
Not the difference
of two squares