1. Polynomials, Factors and Zeros

2. Warm Up
Solve the following by factoring. (Hint: remember to factor out the common monomial first)
x3 – 2x2 – 15x = 0
x3 – x2 – 12x = 0
x3 +8x2 + 16x = 0
4. 2x4 – 32x2 = 0

3. Roots, Zeros and x-intercepts
The following are equivalent about a real number b and a polynomial P(x)
(x – b) is a linear factor of the polynomial P(x)
b is a zero of the polynomial function y = P(x)
b is a root (or solution) of P(x) = 0
4. b is an x-intercept of the graph of y = P(x)

4. Example
What are the zeros of y = x(x – 3)(x + 5)?

5. Try This What are the zeros of the following polynomials?
y = (x + 1)(x – 1)(x – 3)
f(x) = (x + 2)3(x – 4)(x + 5)
g(x) = 5x3 – 10x2
y = x2 + 5x + 6
f(x) = (2x – 3)(5x + 7)(x – 1) 6.

6. Factor Theorem
The expression x – a is a factor of a polynomial if and only if the value a is a zero of the polynomial function.

7. Example
1. Determine whether (x + 5) is a factor of f(x)=3x4 + 15x3 – x If it is write f(x) in factored form.

8. Try This
Show that x + 3 is a factor of f(x) = x4 + 3x3 – x – 3. Then factor f(x) completely.

9. Example
What is the a cubic polynomial function in standard form with zeros -1, 3 and -4?

10. Try This Find the following polynomials in standard form.
A quadratic polynomial function with zeros -3 and 7.
2. A quartic polynomial function with zeros -1, -1, 2 and 3

11. Try This
During the first 5 seconds of a roller coaster ride, the function h(t) = 4t3 – 21t2 + 9t + 34 represents the height h (in feet) of the roller coaster after t seconds. How long is the roller coaster at or below ground level in the first 5 seconds?