1. 6-4 Solving Polynomial Equations
Hubarth
Algebra II

2. Ex. 1 Solve by Graphing
Solve 𝑥 3 +3 𝑥 2 =𝑥+3
Step 1 Graph 𝑦 1 = 𝑥 3 +3 𝑥 2 and 𝑦 2 =𝑥+3
Graphing calculator
Based on the graph. The solutions are −3, −1 𝑎𝑛𝑑 1

3. Ex. 2 Real- World Connection
Pet Transportation The dimensions in inches of a portable kennel can be expressed as width
𝑥, length 𝑥+7 and height 𝑥−1. The volume is 5.9 𝑓𝑡 3 . Find the portable kennels dimensions.
Covert the volume to cubic inches.
5.9 𝑓𝑡 3 ∙ 𝑖𝑛 3 𝑓𝑡 3 = 𝑖𝑛 3
Now graph 𝑦 1 = and 𝑦 2 =𝑥(𝑥+7)(𝑥−1)
Graph
The dimensions are about 27 in. by 20 in. by 19 in.

4. Sum and Difference of Cubes
𝑎 3 + 𝑏 3 = 𝑎+𝑏 𝑎 2 −𝑎𝑏+ 𝑏 2
𝑎 3 − 𝑏 3 =(𝑎−𝑏)( 𝑎 2 +𝑎𝑏+ 𝑏 2 )
Verify the patterns by multiplying. Here are steps for the sum of cubes.
𝑎+𝑏 𝑎 2 −𝑎𝑏+ 𝑏 2 =𝑎 𝑎 2 −𝑎𝑏+ 𝑏 2 +𝑏 𝑎 2 −𝑎𝑏+ 𝑏 2
= 𝑎 3 − 𝑎 2 𝑏+𝑎 𝑏 2 + 𝑎 2 𝑏−𝑎 𝑏 2 + 𝑏 3
= 𝑎 3 + 𝑏 3

5. Ex. 3 Factoring a Sum or Difference of Cubes
Multiple Choice If you factor 𝑥 3 −8 in the form 𝑥−𝐴 𝑥 2 +𝐵𝑥+𝐶 , what is the
value of A?
A. 2 B. -2 C. 4 D. -4
𝑥 3 −8=𝑥 3 −(2 ) 3
=(𝑥−2)( 𝑥 2 +2𝑥+ 2 2 )
=(𝑥−2)( 𝑥 2 +2𝑥+4)
The answer is A, 2

6. Ex. 4 Solving a Polynomial Equation
Solve 27 𝑥 3 +1=0. Find all complex roots.
27 𝑥 3 +1= (3𝑥) 3 + (1) 3
=(3𝑥+1)( 3𝑥 2 −3𝑥+1)
=(3𝑥+1)(9 𝑥 2 −3𝑥+1)
3𝑥+1=0
9𝑥 2 −3𝑥+1 cannot be factor so we
use the quadratic formula.
𝑥= −1 3
𝑎=9, 𝑏=−3, 𝑐=1
𝑥= −(−3)± (−3 ) 2 −4(9)(1) 2(9)
= 3± 9−36 18
= 3± −27 18
= 3±3𝑖
The roots are −1 3 and 1±𝑖 3 6
𝑥= 1±𝑖 3 6

7. Ex. 5 Factoring by Using a Quadratic Pattern
Step 1 Rewrite 𝑥 4 −2 𝑥 2 −8 in the pattern of a quadratic expression, so you can factor it.
Make a temporary substitution of variables.
𝑥 4 −2 𝑥 2 −8= (𝑥 2 ) 2 −2 𝑥 2 −8
= 𝑎 2 −2𝑎−8
Step 2 Factor 𝑎 2 −2𝑎−8
(𝑎−4)(𝑎+2)
Step 3 Substitute back to the original variable and factor if necessary.
𝑎−4 𝑎+2 =( 𝑥 2 −4)( 𝑥 2 +2)
=(𝑥+2)(𝑥−2)( 𝑥 2 +2)

8. Ex. 6 Solving a Higher-Degree Polynomial Equation
Solve 𝑥 4 − 𝑥 2 =12
𝑥 4 − 𝑥 2 −12=0
(𝑥 2 ) 2 − 𝑥 2 −12=0
𝑎 2 −𝑎−12=0
(𝑎−4)(𝑎+3)
( 𝑥 2 −4)( 𝑥 2 +3)
(𝑥+2)(𝑥−2)( 𝑥 2 +3)
𝑥+2= 𝑥−2= 𝑥 2 +3=0
𝑥=− 𝑥= 𝑥 2 =−3
x=i 3

9. Practice
1. Graph and solve 𝑥 3 −2 𝑥 2 =−3
graph
2. Factor 8𝑥 3 −1
(2𝑥−1)(4 𝑥 2 +2𝑥+1)
3. Solve 𝑥 3 +8=0
−2, 1±𝑖 3
4. Factor 𝑥 4 +7 𝑥 2 +6
( 𝑥 2 +6)( 𝑥 2 +1)
5. Solve 𝑥 𝑥 2 +18=0
3𝑖, −3𝑖, 𝑖 2 , −𝑖 2