Download Factoring Special Polynomials

LESSON 7: FACTORING SPECIAL POLYNOMIALS
Learning Outcome: Learn to investigate some special factoring patterns.
With a partner, determine each product:
(x + 1)²
(x + 2)²
(x + 3)²
(x - 1)²
(x – 2)²
(x – 3)²
(2x + 1)²
(3x + 1)²
(4x + 1)²
(2x – 1)²
(3x – 1)²
(4x – 1)²
What patterns do you see in the trinomials and their factors above?
How could you use the patterns to factor these trinomials?
4x² + 20x + 25
9x² - 12x + 4
Write a strategy for factoring polynomials of this type.
When given the situation of  x  2  . The term “squaring” means multiplying
the number by itself.
2
Ex.  x  4    x  4  x  4  
2
Ex.  3x  5   3x  5 3x  5 
2
Ex.     
2
General expansion of squared binomials or Perfect Square Trinomials:
 a  b   a 2  2ab  b2
2
 a  b   a 2  2ab  b2
2
Factoring Perfect Square Trinomials:
Patterns:  a  b   a 2  2ab  b 2 and
2
a  b
2
 a 2  2ab  b 2
Factoring PST:
4 x 2  20 x  25
- recognize it first
how recognize: 1 and last term are perfect squares
st
a  4x2  4x2  2x
c  25  25  5
b  2ab  2  2 x  5   20 x
It works, all the
numbers match and
add up correctly.
Now to factor:
4 x 2  20 x  25
square root the 1st and last terms: a = 2x, c = 5
To write in form pay attention to the sign in front of the b value:
(in this case +)
2
Write in form:  2 x  5 
Ex. Factor :
a. 25a 2  20a  4
b. 4a 2  4ab  b2
Difference of Squares:
Ex.  x  1 x 1  x2  x  x 1  x2 1
Ex. 3x  53x  5  9 x2 15x  15x  25  9 x2  25
Ex.         2     2  2 
2
Notice that the two terms are
subtracted and each term is
squared (Difference of squares)
General Rule:
 a  b a  b  a2  b2
Factoring Difference of Squares:
Pattern: a2  b2   a  b  a  b 
Factoring DOS:
9 x 2  16
9x
Ex. Factor each:
a. 9 x 2  25
2
- recognize it first
16 - square root the 1st and last terms
3x
4
Write in form: (3x-4)(3x+4)
b. 100 y 2  9 x 2
c. 16m2  4n2
d. 162𝑣 4 − 2𝑤 4
Assignment: pg. 194-195 #4-8, 10-13, 15, 18, 20