The document discusses factoring polynomials using the difference of squares pattern and perfect square trinomial pattern. It provides examples of factoring expressions involving these patterns, including solving polynomial equations. The examples illustrate factoring expressions with variables and evaluating expressions to solve vertical motion problems.

2. Difference of Two Squares (Factor)
a - b = (a + b)(a - b)
2
2
y -16 = y - 4 = (y + 4)(y - 4)
2
2
2

3. Example 1
Factor the difference of two squares
Factor the polynomial.
a. 25m2 – 36 = ( 5m )2 – 62
Write as a2 – b2.
= (5m + 6 ) (5m – 6 ) Difference of two squares
pattern
b. x2 – 49y2 = x2 – ( 7y)2
= (x + 7y ) (x – 7y)
c. 8 – 18n2 = 2( 4 – 9n2 )
= 2[22 – ( 3n)2]
Write as a2 – b2.
Difference of two squares
pattern
Factor out common factor.
Write 4 – 9n2 as a2 – b2.

4. Example 1
Factor the difference of two squares
= 2(2 + 3n ) (2 – 3n)
d. – 9 + 4x2 = 4x2 – 9
Difference of two squares
pattern
Rewrite as difference.
= ( 2x )2 – 32
Write as a2 – b2.
= (2x + 3 ) (2x – 3 )
Difference of two squares
pattern

5. Perfect Square Trinomial
x 2 + 6x + 9
a + 2ab + b
2
x 2 -10x + 25
2
a - 2ab + b
2
2

6. Perfect Square Trinomial (Factor)
a2 + 2ab + b2 = (a + b)2
Algebra
Example
x + 6x + 9 = x + 2(x ×3) + 3 = (x + 3)
Algebra
(a - b)2
a - 2ab + b =
2
2
2
2
2
2
(x - 5)2
Example x -10x + 25 = x - 2(x × 5)+ 5 =
2
2
2

7. Example 2
Factor perfect square trinomials
Factor the polynomial.
a. n2 – 12n + 36 = n2 – 2( n • 6) + 62
= (n – 6 )2
Write as a2 – 2ab + b2.
Perfect square
trinomial pattern
b. 9x2 – 12x + 4 = ( 3x)2 – 2( 3x • 2) + 22 Write as a2 – 2ab + b2.
= (3x – 2 )2
Perfect square
trinomial pattern
c. 4s2 + 4st + t2 = ( 2s )2 + 2( 2s • t ) + t2 Write as a2 + 2ab + b2.
= (2s + t )2
Perfect square
trinomial pattern

8. Example 3
Multiple Choice Practice
What is the factored form of – 3x2 + 36xy – 108y2 ?
( – 3x – 6y )2
– 3( x + 6y )2
( 3x – 6y )2
– 3( x – 6y )2
SOLUTION
–3x2 + 36xy – 108y2 = –3( x2 – 12xy + 36y2 )
Factor out – 3 .
2
= –3[x2 – 2 (x • 6y ) + ( 6y) ]
Write x2 – 12xy + 36y2 as a2 – 2ab + b2.

9. Example 3
Multiple Choice Practice
= – 3( x – 6y )2
ANSWER
Perfect square trinomial pattern
The correct answer is D.

10. Example 4
Solve a polynomial equation
Solve the equation
x2
x2
1
x +
+
= 0.
3
9
= 0
Write original equation.
6x + 1 = 0
Multiply each side by 9.
2
1
x +
+
3
9x2 +
2
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
x = –
Zero-product property
1
3
Solve for x.

11. Example 4
ANSWER
Solve a polynomial equation
The solution of the equation is
–
1
3
.

12. Example 5
Solve a vertical motion problem
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.
Because the sponge was dropped, it has no initial
vertical velocity. To determine when the sponge lands
on the ground, find the value of t for which the height is
0.

13. Example 5
Solve a vertical motion problem
h = – 16t2 + vt + s
Vertical motion model
0 = – 16t2 + ( 0 ) t + 64
Substitute 0 for h, 0 for v, and
64 for s.
0 = – 16 ( t2 – 4 )
Factor out – 16 .
0 = – 16 ( t + 2 ) ( t – 2 )
Difference of two squares pattern
t + 2 = 0
or
t = –2
or
t – 2 = 0
t = 2
Zero-product property
Solve for t.
Disregard the negative solution of the equation.

14. Example 5
Solve a vertical motion problem
ANSWER
The sponge lands on the ground 2 seconds after it is
dropped.

15. 9.8 Warm-Up (Day 1)
Factor the polynomial.
1.
x2 - 9
2.
4y 2 - 64
3.
h + 4h + 4
4.
2y2 - 20y + 50
2

16. 9.8 Warm-Up (Day 2)
Solve the equation.
1.
a2 + 6a + 9 = 0
2.
w -14w + 49 = 0
3.
n -81= 0
2
2