Algebra 2
5.4 notes – part 1: Factor and Solve Polynomial Equations
Warm-Up:
Factor the following quadratic expressions (check for GCF, grouping)
1. x 2  5 x  6
2. 2 x 2  14 x  60
3. x 2  16
4. x 2  49
5. 6 x 2  11x  3
6. 3 x 2  11x  20
Find the degree of each polynomial, then factor each polynomial.
1. x 3  5 x 2  6 x
2. x 3  4 x 2  2 x  8
Degree = ______
Degree = ______
3. x 3  9 x
4. x 3  x 2  2 x  2
Degree = ______
Degree = ______
5. x 3  8 x 2  15 x
6. x 3  6 x 2  8 x
Degree = ______
Degree = ______
7. x 3  2 x 2  14 x  7 x 2
3
2
8. 18 x  60 x  50 x
Degree = ______
Degree = ______
9. x 3  3x 2  4 x  12
3
2
10. x  x  2x  2
Degree = ______
Degree = ______

Factor: Sum and Difference of 2 cubes
Example:
Sum: x 3  27


Sum: 27 x 3  64
Difference: 8 x 3  125
Practice: Factor each polynomial.
1. x 3  1000

2. x 3 125

3. x 3  64


Difference: x 3 1
4. x 3  8

5. x 6  125
6. 27x 3 125

Solving Polynomial Equations!
Use factoring to solve each polynomial equation.
STEPS: 1. Set the polynomial equal to 0, and factor.
2. Set each factor equal to 0 and solve the mini-equations.
2 x3  7 x 2  3x  0
1. x 2  2 x  3  0
2.
3. x 3  x 2  2 x  2  0
4. 3x3  6 x 2  3x  0
5. x 3  x 2  x  1  0
6. x 3  5 x 2  9 x  45  0
Architecture
You are designing a marble basin that will hold a fountain for a city park. The basin’s sides and bottom should be 1 foot
thick. Its outer length should be twice its outer width and outer height.
What should the outer dimensions of the basin be if it is to hold 36 cubic feet of water?
1 ft
Interior Length • Interior Width • Interior Height = Volume
x
x
2x
Archaeology
At the ruins of Caesarea, archaeologists discovered a huge hydraulic concrete block with a volume of 945 cubic meters.
The blocks dimensions are x meters high, by 12x – 15 meters long by 12x – 21 meters wide. What is the height of the
block?
Volume of a rectangular prism = length ∙ width ∙ height
Algebra 2
5.4 notes – part 2: Factor and Solve Polynomial Equations
Solving Polynomial equations by factoring AND by graphing
1. Solve by factoring: x 3  8 x 2  12 x  0
Now graph f ( x)  x 3  8 x 2  12 x
2. Solve 2 x 3  3 x 2  2 x  0
Now graph f ( x)  2 x 3  3x 2  2 x
These polynomial functions have three factors and three solutions.
These solutions are the __________________________ , also known as the ____________.
Find all the roots of f ( x)  x3  2 x 2  9 x  18 .
First Method: Solving by hand!
Second Method: Graphing!
Find all the roots of f ( x)  x 4  4 x 2  3
First Method: Solving by hand!
Second Method: Graphing!