Chapter 9.5 Notes: Solve Polynomial Equations in

Factored Form

Goal: You will solve polynomial equations.

• 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

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. 6x5; 30x4 i.e. 45x4y; 60x5y2

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. 4x4 + 24x3

Ex.6: Factor out the greatest common monomial factor.

a. 8x + 12y

b. 14y2 + 21y

Ex.7: Solve the equation by factoring.

a. Solve 2x2 = -8x

b. Solve 3x2 = -18x

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 6x2 – 15x.

Ex.9: Find the roots of 4s2 – 14s.

Ex.10: Solve 3s2 – 9s = 0.

Ex.11: Find the roots of a2 + 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.

• Vertical Motion Model:The height h (in feet) of a projectile can be modeled by

h = -16t2 + 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?