1. Solutions of Polynomial Equations
Section 6.3
Solutions of Polynomial Equations
Copyright ©2013 Pearson Education, Inc.

2. Objectives
Solve polynomial equations using factoring, factoring by grouping, and the root method
Find factors, zeros, x-intercepts, and solutions
Estimate solutions with technology
Solve polynomial equations using the intersection method and the x-intercept method

3. Solving Polynomial Equations by Factoring
The following statements regarding a function f are equivalent: (x  a) is a factor of f(x). a is a zero of the function f. a is a solution to the equation f(x)= 0. a is an x-intercept of the graph of y = f(x). The graph crosses the x-axis at the point (a, 0).

4. Example
If find a. the zeros of P(x). b. the solutions of P(x) = 0. c. the x-intercepts of the graph of y = P(x). Solution a. The factors of P(x) are (x – 1), (x + 2), and (x + 5); thus the zeros are x = 1, x = –2, and x = –5. b. The solutions are the zeros x = 1, x = –2, and x = –5. c. The x-intercepts are 1, –2, and –5.

5. Example
The amount y of photosynthesis that takes place in a certain plant depends on the intensity x of the light (in lumens) present, according to the function y = 120x2 - 20x3. The model is valid only for nonnegative x-values that produce a positive amount of photosynthesis. a. Graph this function with a viewing window large enough to see 2 turning points. b. Use factoring to find the x-intercepts of the graph. c. What nonnegative values of x will give positive photosynthesis? d. Graph the function with a viewing window that contains only the values of x found in part (c) and nonnegative values of y.

6. Example (cont)
Solution a. The graph shows the function with two turning points. b. Factor to find the intercepts. c. Positive output values result when input values are between 0 and 6. Thus, the nonnegative x-values that produce positive amounts of photosynthesis are 0 < x < 6.

7. Example (cont)
Solution d. Graphing the function with a viewing window [0,6] by [0, 700] will show the region that is appropriate for the model.

9. Example
Solve the following equation. a. x3 = 125 b. 5x4 = 80 c. 4x2 = 18 Solution a. b. c.

10. Example
The future value of $10,000 invested for 4 years at interest rate r, compounded annually, is given by S = 10,000(1 + r)4. Find the rate r, as a percent, for which the future value is $14,641. Solution Solve
Negative value cannot be an interest rate.
r = 0.1 = 10%

11. Example (cont)
The graphs support the conclusion.

12. Example
The annual number of arrests for crime per 100,000 juveniles from 10 to 17 years of age can be modeled by the function
where x is the number of years after 1980.† The number of arrests per 100,000 peaked in 1994 and then decreased. Use this model to estimate graphically the year after 1980 in which the number of arrests fell to 234 per 100,000 juveniles.
† Note that some juveniles have multiple arrests.

13. Example (cont)
Solve the equation: Use the intersection method. The point where the graphs intersect is about 20. The number of arrests fell to 234 per 100,000 during the year 2000.