1. Using Factoring To Solve
Section 8.5
Using Factoring To Solve
Polynomial Equations

2. Zero Factor Property Property Let A and B be real number:
If AB = 0, then A = 0 or B = 0.
In words: If the product of two numbers is zero, then at least one of the numbers must be zero.

3. Solving Quadratic Equations in One Variable
Solving a Quadratic Equation
Example
Solve (x – 5)(x + 2) = 0.

4. Solving Quadratic Equations in One Variable
Solving a Quadratic Equation
Solution

5. Solving Quadratic Equations in One Variable
Solving a Quadratic Equation
Example
Solve w2 – 2w – 8 = 0.

6. Solving Quadratic Equations in One Variable
Solving a Quadratic Equation
Solution

7. Find the x-intercepts of the graph of f(x) = x2 – 7x + 10.
Connection Between x-intercepts of a Quadratic Function
Finding x-intercepts of a Quadratic Function
Example
Find the x-intercepts of the graph of f(x) = x2 – 7x + 10.

8. Check with a graphing calculator
Connection Between x-intercepts of a Quadratic Function
Finding x-intercepts of a Quadratic Function
Solution
Check with a graphing calculator

9. Solving More Quadratic Equations
Solving Quadratic Equations
Example
Solve.
1. 4x2 + 26x + 30 = x2 – 10x + 25 = 0.

10. Solving More Quadratic Equations
Solving Quadratic Equations
Solution 1.

11. Solving More Quadratic Equations
Solving Quadratic Equations
Solution Continued 2.

12. Solving More Quadratic Equations
Solving Quadratic Equations
Rule
The solution set of a quadratic equation in one variable may contain two real numbers, one real number, or no real numbers.

13. Solving More Quadratic Equations
Solving Quadratic Equations
Example
Solve 2x2 – 8x = 5x – 20.

14. Solving More Quadratic Equations
Solving Quadratic Equations
Solution

15. Solving More Quadratic Equations
Solving a Quadratic Equation That Contains Fractions
Example
Solve 1/2t2 + 1/3 = 5/6t

16. Solving More Quadratic Equations
Solving a Quadratic Equation That Contains Fractions
Solution

17. Solving More Quadratic Equations
Solving a Quadratic Equation That Contains Fractions
Solution Continued
Example

18. Cubic Equations in One Variables
Solving a Cubic Equation
Example
Solve 2x3 = 42x + 8x2.

19. Cubic Equations in One Variables
Solving a Cubic Equation
Solution

20. Cubic Equations in One Variables
Solving a Cubic Equation
Process
If an equation can be solved by factoring, we solve it
by the following steps:
1. Write the equation so that one side of the equation is equal to zero.
2. Factor the nonzero side of the equation.
3. Apply the zero factor property.
4. Solve each equation that results from applying the zero factor property.

21. Find the x-intercept of the graph of f(x) = x3 – 5x2 – 4x + 20.
Cubic Equations in One Variables
Finding x-intercepts of the graph of a Cubic Function
Example
Find the x-intercept of the graph of
f(x) = x3 – 5x2 – 4x + 20.
Solution

22. Use graphing to solve x2 – x – 7 = – x2.
Solving Polynomial Equations in One Variable by Graphing
Using Graphing to Solve an Equation in One Variable
Example
Use graphing to solve x2 – x – 7 = – x2.

23. Solution Use “intersect” on a graphing calculator to find solutions
Solving Polynomial Equations in One Variable by Graphing
Using Graphing to Solve an Equation in One Variable
Solution
Use “intersect” on a graphing calculator to find solutions
Approx. Solution (–1.64, –2.68) and (2.14, –4.57)