1. Factor and Solve Polynomial Equations Homework Questions?
Chapter 5.4
Factor and Solve Polynomial Equations
Homework Questions?

2. Warm up! Multiply the polynomials 𝑥+2 𝑥+3 (2𝑥−1)(2𝑥+1)
𝑥+2 𝑥 (2𝑥−1)(2𝑥+1)
(𝑥−7 ) 𝑥 2 (𝑥+5)

3. Special Case Polynomials Factoring Polynomials
Difference of Squares
𝒂 𝟐 − 𝒃 𝟐 =(𝒂−𝒃)(𝒂+𝒃)
𝟗 𝒙 𝟐 −𝟏= 𝟑𝒙−𝟏 𝟑𝒙+𝟏
Sum and Difference of Cubes
𝒂 𝟑 + 𝒃 𝟑 =(𝒂+𝒃)( 𝒂 𝟐 −𝒂𝒃+ 𝒃 𝟐 )
𝟖 𝒙 𝟑 +𝟐𝟕=(𝟐𝒙+𝟑)(𝟒 𝒙 𝟐 −𝟔𝒙+𝟗)
𝒂 𝟑 − 𝒃 𝟑 =(𝒂−𝒃)( 𝒂 𝟐 +𝒂𝒃+ 𝒃 𝟐 )
𝟔𝟒 𝒙 𝟑 −𝟏=(𝟒𝒙−𝟏)(𝟏𝟔 𝒙 𝟐 −𝟒𝒙+𝟏)
Binomial Square (Perfect Square Trinomials)
𝒂 𝟐 +𝟐𝒂𝒃+ 𝒃 𝟐 =(𝒂+𝒃 ) 𝟐
𝒙 𝟐 +𝟔𝒙+𝟗=(𝒙+𝟑 ) 𝟐
𝒂 𝟐 −𝟐𝒂𝒃+ 𝒃 𝟐 =(𝒂−𝒃 ) 𝟐
𝟒 𝒙 𝟐 −𝟏𝟐𝒙+𝟗=(𝟐𝒙−𝟑 ) 𝟐
Binomial Cubes (Perfect Cube Polynomials)
𝒂 𝟑 +𝟑 𝒂 𝟐 𝒃+ 𝟑𝒂𝒃 𝟐 + 𝒃 𝟑 =(𝒂+𝒃 ) 𝟑
𝒙 𝟑 +𝟔 𝒙 𝟐 +𝟏𝟐𝒙+𝟖=(𝒙+𝟐 ) 𝟑
𝒂 𝟑 −𝟑 𝒂 𝟐 𝒃+ 𝟑𝒂𝒃 𝟐 − 𝒃 𝟑 =(𝒂−𝒃 ) 𝟑
𝒙 𝟑 −𝟗 𝒙 𝟐 +𝟐𝟕𝒙−𝟐𝟕=(𝒙−𝟑 ) 𝟑
Factoring (Grouping)
𝒙 𝟐 +𝟓𝒙+𝟒=(𝒙+𝟒)(𝒙+𝟏)
𝟐𝒙 𝟐 −𝟑𝒙−𝟐𝟎=(𝟐𝒙+𝟓)(𝒙−𝟒)
Common Factor (Factor Out)
𝒙 𝟐 +𝒙=𝒙(𝒙+1)
𝟖𝒙 𝟐 +𝟐𝟎𝒙=𝟒𝒙(𝟐𝒙+𝟓)

4. Sum and Difference of Cubes
Factoring with Sum and Difference of Cubes
Sum and Difference of Cubes
𝒂 𝟑 + 𝒃 𝟑 =(𝒂+𝒃)( 𝒂 𝟐 −𝒂𝒃+ 𝒃 𝟐 )
𝟖 𝒙 𝟑 +𝟐𝟕=(𝟐𝒙+𝟑)(𝟒 𝒙 𝟐 −𝟔𝒙+𝟗)
𝒂 𝟑 − 𝒃 𝟑 =(𝒂−𝒃)( 𝒂 𝟐 +𝒂𝒃+ 𝒃 𝟐 )
𝟔𝟒 𝒙 𝟑 −𝟏=(𝟒𝒙−𝟏)(𝟏𝟔 𝒙 𝟐 −𝟒𝒙+𝟏)
𝑥 3 +64=
8𝑥 3 +27
𝑥 3 −64=
16𝑥 5 −250 𝑥 2

5. X box (i.e. products of c, sums of b)
Factoring Flow Chart
Is it a polynomial?
Can you factor
out a GCF?
Type of
Polynomial?
Factor out!
Monomial
Fin!
Put it standard
from
Fin!
Binomial
Trinomial
Terms >3
Sum
Difference
Squares
Not Factorable
Difference of
squares
X box (i.e. products of c, sums of b)
Not Factorable
Factor
a=1
Factor by Grouping
Not Factorable
Factor
Sum
Difference
Cubes
Sum of Cubes
Difference of
a≠1
Factor by Grouping
Not Factorable
Factor
Always Check Your Solution by Multiplying the Solution (i.e. FOIL)

6. Factoring by Grouping when # terms >3
𝑥 3 − 3𝑥 2 −16𝑥+48
27𝑥 𝑥 2 −3𝑥−5
(𝑥 3 − 3𝑥 2 )+(−16𝑥+48)
(27𝑥 𝑥 2 )+(−3𝑥−5)
𝑥 2 (𝑥−3)−16(𝑥−3)
9𝑥 2 (3𝑥+5)−1(3𝑥+5)
(𝑥 2 −16)(𝑥−3)
(9𝑥 2 −1)(3𝑥+5)
(𝑥−4)(𝑥+4)(𝑥−3)
(3𝑥−1)(3𝑥+1)(3𝑥+5)

7. Factoring the following completely
2𝑥 3 − 𝑥 2 +2𝑥−1
4𝑥 𝑥 2 −𝑥−4
𝑥 3 +64
4𝑥 6 −20 𝑥 𝑥 2
𝑥 3 +7 𝑥 2 −9𝑥−63
16 𝑥 4 −625