Transcript Document

9.7 Factor Special Products
Warm Up
Lesson Presentation
Lesson Quiz
9.7
Warm-Up
Find the product.
1. (m + 2)(m – 2)
ANSWER
m2 – 4
2. (2y – 3)2
ANSWER
4y2 – 12y + 9
3. (s + 2t)(s – 2t)
ANSWER
s2 – 4t2
9.7
Warm-Up
4. A football is thrown in the air at an initial height of
5 feet and an initial velocity of 16 feet per second.
After how many seconds does it hit the ground?
ANSWER
1.25 sec
9.7
Example 1
Factor the polynomial.
a. y2 – 16 = y2 – 42
= (y + 4)(y – 4)
b.
25m2 – 36 = (5m)2 – 62
= (5m + 6)(5m – 6)
c. x2 – 49y2 = x2 – (7y)2
= (x + 7y)(x – 7y)
Write as a2 – b2.
Difference of two squares
pattern
Write as a2 – b2.
Difference of two squares
pattern
Write as a2 – b2.
Difference of two squares
pattern
9.7
Example 2
Factor the polynomial 8 – 18n2.
8 – 18n2 = 2(4 – 9n2)
Factor out common factor.
= 2[22 – (3n) 2]
Write 4 – 9n2 as a2 – b2.
= 2(2 + 3n)(2 – 3n)
Difference of two squares pattern
9.7
Guided Practice
1. Factor the polynomial 4y2 – 64 .
ANSWER
(2y + 8)(2y – 8)
9.7
Example 3
Factor the polynomial.
a.
n2 – 12n + 36 = n2 – 2(n 6) + 62
= (n – 6)2
b.
9x2 – 12x + 4 = (3x)2 – 2(3x  2) + 22
= (3x – 2)2
c.
4s2 + 4st + t2 = (2s)2 + 2(2s t) + t2
= (2s + t)2
Write as a2 – 2ab + b2.
Perfect square
trinomial pattern
Write as a2 – 2ab + b2.
Perfect square
trinomial pattern
Write as a2 + 2ab + b2.
Perfect square
trinomial pattern
9.7
Example 4
Factor the polynomial –3y2 + 36y – 108.
–3y2 + 36y – 108 = –3(y2 – 12y + 36)
Factor out –3.
= –3[y2 – 2(y 6) + 62] Write y2 – 12y + 36
as a2 – 2ab + b2.
= –3(y – 6)2
Perfect square
trinomial pattern
9.7
Example 4
CHECK
Check your factorization
using a graphing calculator.
Graph y1 = –3y2 + 36y – 108
and y2 = –3(y – 6)2. Because
the graphs coincide, you
know that your factorization
is correct.
9.7
Guided Practice
Factor the polynomial.
2.
h2 + 4h + 4 = (h + 2)2
3.
2y2 – 20y + 50 = 2(y – 5)2
4.
3x2 + 6xy + 3y2 = 3(x + y)2
9.7
Example 5
2
1
Solve the equation x2 + 3 x + 9 = 0.
2x 1
2
x + 3 +9=0
Write original equation.
9x2 + 6x + 1 = 0
Multiply each side by 9.
(3x)2 + 2(3x 1) + (1)2 = 0
Write left side as a2 + 2ab + b2.
(3x + 1)2 = 0
Perfect square trinomial pattern
3x + 1 = 0
ANSWER
x=–
Zero-product property
1
3
Solve for x.
The solution of the equation is – 1 .
3
9.7
Guided Practice
Solve the equation
5.
a2 + 6a + 9 = 0
6.
w2 – 14w + 49 = 0
7.
n2 – 81= 0
a = –3
w=7
n = – 9 or n = 9
9.7
Example 6
FALLING OBJECT
A window washer drops a wet
sponge from a height of 64 feet.
After how many seconds does
the sponge land on the ground?
SOLUTION
Use the vertical motion model to write an
equation for the height h (in feet) of the sponge
as a function of the time t (in seconds) after it is
dropped.
9.7
Example 6
The sponge was dropped, so it has no initial vertical
velocity. Find the value of t for which the height is 0.
h = –16t2 + vt + s
0 = –16t2 + (0)t + 64
0 = –16(t2 – 4)
Substitute 0 for h, 0 for v, and 64 for s.
0 = –16(t – 2)(t +2)
Difference of two squares pattern
t – 2 = 0 or t + 2 = 0
t = 2 or t = –2
Zero-product property
Vertical motion model
Factor out –16.
Solve for t.
Disregard the negative solution of the equation.
ANSWER
The sponge lands on the ground 2
seconds after it is dropped.
9.7
8.
Guided Practice
WHAT IF? In Example 6, suppose the sponge is
dropped from a height of 16 feet. After how many
seconds does it land on the ground?
ANSWER
The sponge lands on the ground 1
second after it is dropped.
9.7
Lesson Quiz
Factor the trinomial.
1.
4m2 – n2
ANSWER
(2m – n)(2m + n)
2. x2 + 6x + 9
ANSWER
3.
(x + 3)2
4y2 – 16y +16
ANSWER
(2y – 4)2
9.7
4.
Lesson Quiz
Solve the equation
1
+x+
=0
4
1
–
ANSWER
2
x2
5.
An apple falls from a branch 9 feet above the
ground. After how many seconds does the apple
hit the ground?
ANSWER
0.75 sec