Applications of Polynomial Equation

1. Applications of Polynomial Equation Objectives: The Principle of Zero Products Problem Solving

2. Whenever two polynomials are set equal to each other, we have a polynomial equation. Some examples of polynomial equations are: The degree of a polynomial equation is the same as the highest degree of any term in the equation. Thus, from left to right, the degree of each equation listed above is 3, 2, and 4

3. The Principle of Zero Products For any real numbers a and b:if ab = 0, then a = 0 or b = 0. If a = 0 or b = 0, then ab = 0

4. Solve: To apply the principle of zero products, we need 0 on one side of the equation. Thus subtract 6 from both sides: Getting 0 on one side To express the polynomial as a product, we factor Factoring The principle of zero products says that since (x-3)(x+2) is 0, then Using the principle of zero products Each of these linear equations is then solved separately:

5. We check as follows: Both 3 and -2 are solutions. The solution set is {-2, 3}

6. To Use the Principle of Zero Products • Write an equivalent equation with 0 on one side, using the addition principle. • Factor the nonzero side of the equation. • Set each factor that is not a constant equal to 0 • Solve the resulting equations.

7. Let’s try: Solve:

8. Given that f(x)= find all values of a for which f(a) = 4 We want all numbers a for which f(a) = 4. Since Setting f(a) equal to 4 Getting 0 on one side Factoring Check:

9. Given that f(x)= and g(x) = find all x-values for which f(x) = g(x) We substitute the polynomial expressions for f(x) and g(x) and solve the resulting equation: Substituting Getting 0 on one side and writing in descending order. Factoring out a common factor Factoring the trinomial Using the principle of zero products Check:

10. Find the domain of F if F(x) = The domain of F is the set of all values for which F(x) Is a real number. Since division by 0 is undefined, F(x) cannot be calculated for any x-value for which the denominator, is 0. To made sure these values are excluded. Setting the denominator equal to 0 Factoring These are the values to exclude. The domain of F is {x| x is a real number and x ≠ -5 and x ≠ 3}

11. Problem Solving During intermission at sporting events, it has become common for team mascots to use a powerful slingshot to launch tightly rolled tee shirts into the stands. The height h(t), in feet, of an airborne tee shirt t seconds after being launched can be approximated by After peaking, a rolled-up tee shirt is caught by a fan 70 ft. above ground level. How long was the tee shirt in the air?

12. Familiarize We make a drawing and label it, using the information provided (see figure to the left). If we wanted to, we could evaluate h(t) for a few values of t. Note the t cannot be negative, since it represents time from launch.

13. Translate The relevant function has been provided. Since we are asked to determine how long it will take for the shirt to reach someone 70ft. Above ground level, we are interested in the value of t for which h(t) = 70.

14. Carry out We solve the quadratic equation: Subtracting 70 from both sides Factoring

15. Check We have: Both 4 and 1 check. However, the problem states that the tee shirt is caught after peaking. Thus we reject 1 since that would indicate when the height of the tee shirt was 70 ft on the way up

16. State The tee shirt was in the air for 4 sec. before being caught 70 ft above ground.