1. Chapter 9.5 Notes: Solve Polynomial Equations in Factored Form Goal: You will solve polynomial equations.

2. Zero-Product Property: Let a and b be real numbers. If ab = 0, then a = 0 or b = 0. i.e. (x + 8)(x – 3) = 0 i.e. 2x(x – 10) = 0 Ex.1: Solve (x – 4)(x + 2) = 0 Ex.2: Solve (x – 5)(x – 1) = 0 Ex.3: Solve (2x + 5)(3x – 2) = 0

3. Ex.4: Solve 8x(3x – 6) = 0 Greatest Common Factor (GCF) The Greatest Common Factor of two or more whole numbers is the largest whole number that divides evenly into each of the numbers. The Greatest Common Factor of two or more same variable terms with exponents is the lowest exponent that goes into each of the exponents with the same variable. i.e. 14; 24 i.e. 6x 5 ; 30x 4 i.e. 45x 4 y; 60x 5 y 2

4. Factoring To solve a polynomial equation using the zero- product property, you may need to factor the polynomial, which involves writing it as a product of other polynomials. One step in factoring is to look for the greatest common monomial factor of the polynomial’s terms. Ex.5: Factor out the greatest common monomial factor. a. 12x + 42y b. 4x 4 + 24x 3

5. Ex.6: Factor out the greatest common monomial factor. a. 8x + 12y b. 14y 2 + 21y Ex.7: Solve the equation by factoring. a. Solve 2x 2 = -8x b. Solve 3x 2 = -18x

6. Roots A root of a polynomial involving x is a value of x for which the corresponding value of the polynomial is 0. – Roots means the same thing as solutions. Ex.8: Find the roots of 6x 2 – 15x. Ex.9: Find the roots of 4s 2 – 14s. Ex.10: Solve 3s 2 – 9s = 0.

7. Ex.11: Find the roots of a 2 + 5a. Vertical Motion A projectile is an object that is propelled into the air but has no power to keep itself in the air. – A thrown ball is a projectile, but an airplane is not. The height of a projectile can be described by the vertical motion model.

8. Vertical Motion Model: The height h (in feet) of a projectile can be modeled by h = -16t 2 + vt + s where t is the time (in seconds) the object has been in the air, v is the initial vertical velocity (in feet per second), and s is the initial height (in feet). Ex.12: As a salmon swims upstream, it leaps into the air with an initial vertical velocity of 10 feet per second. After how many seconds does the salmon return to the water?