Sec 3.2 – Polynomial Functions
Factoring Using Polynomial Identities
Name:
Common Polynomial Identities:
Description
Identity
Example
Difference of Two Squares
𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏)
9𝑥 2 − 4𝑦 2 = (3𝑥 + 2𝑦)(3𝑥 − 2𝑦)
Sum of Two Squares
𝑎2 + 𝑏 2 = (𝑎 + 𝑏𝑖)(𝑎 − 𝑏𝑖)
16𝑚2 + 9 = (4𝑚 + 3𝑖)(4𝑚 − 3𝑖)
Perfect Square Trinomial
𝑎2 + 2𝑎𝑏 + 𝑏 2 = (𝑎 + 𝑏)2
9𝑎2 + 24𝑎 + 16 = (3𝑎 + 4)2
Perfect Square Trinomial
𝑎2 − 2𝑎𝑏 + 𝑏 2 = (𝑎 − 𝑏)2
25𝑝2 − 30𝑝𝑞 + 9𝑞 2 = (5𝑝 + 3𝑞)2
Binomial Cubed
𝑎3 + 3𝑎2 𝑏 + 3𝑎𝑏 2 + 𝑏 3 = (𝑎 + 𝑏)3
𝑥 3 + 6𝑥 2 + 12𝑥 + 8 = (𝑥 + 2)3
Binomial Cubed
𝑎3 − 3𝑎2 𝑏 + 3𝑎𝑏 2 − 𝑏 3 = (𝑎 − 𝑏)3
𝑏 3 − 9𝑏 2 + 27𝑏 − 27 = (𝑏 − 3)3
Difference of Two Cubes
𝑎3 − 𝑏 3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 )
8𝑤 3 − 27 = (2𝑤 − 3)(4𝑤 2 + 6𝑤 + 9)
Sum of Two Cubes
𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 )
64𝑦 3 + 1 = (4𝑦 + 1)(16𝑦 2 − 4𝑦 + 1)
1. Factor the following using a Difference or Sum of Two Squares.
a.
4𝑎2 − 25𝑏 2
d. 𝑥 2 + 36
b. (5𝑚3 )2 − (6𝑛)2
c. 𝑎2 𝑏 8 − 9𝑝6 𝑞 2
e. 18𝑝2 − 98𝑞 2
f. 16𝑤 2 + 7
M. Winking
Unit 3-2 page 45
2. Factor the following using a Difference or Sum of Two Cubes.
𝑎3 − 64
a.
c. 16𝑚9 − 250
b. 27𝑥 3 + 8𝑦 6
d. (7𝑝2 )3 + (2𝑞 4 )3
3. Verify the following polynomial identity from each side.
a.
(𝑎2 + 𝑏 2 )2 = (𝑎2 − 𝑏 2 )2 + (2𝑎𝑏)2
b. (𝑎2 + 𝑏 2 )2 = (𝑎2 − 𝑏 2 )2 + (2𝑎𝑏)2
c. This specific identity is commonly used to find sets of Pythagorean triples.
i.
Find the Pythagorean triple that would be created by using a = 3 and b = 2.
ii.
Find the Pythagorean triple that would be created by using a = 5 and b = 2.
M. Winking
Unit 3-2 page 46
4. Verify the following polynomial identity.
𝑥 2 + 𝑦 2 + 𝑧 2 + 2(𝑥𝑦 + 𝑥𝑧 + 𝑦𝑧) = (𝑥 + 𝑦 + 𝑧)2
M. Winking
Unit 3-2 page 47