 # Finding Rational Zeros.

## Presentation on theme: "Finding Rational Zeros."— Presentation transcript:

Finding Rational Zeros

Zeros = Solutions = Roots = x-intercepts
Find all zeros of x2 –10x + 24 You just factor and set each factor to zero. (You knew this already!) (x – 12)(x + 2) = 0 x = 12, -2 You could also graph and see where it crosses the x-axis (x-intercepts) (You knew this already too!)

Refresh. What is a Rational Number?
A number that can be written as the ratio of two integers. Examples: It can also be an ending or repeating decimal. Examples: … …

= Rational Zero Theorem
If f(x) = anxn + … + a1x + a0 (it’s a polynomial) and the polynomial has integer coefficients, then EVERY rational zero of f has the following form: = factor of the constant term . factor of the leading coefficient

= p is all of the factors of the constant term. 1, 3, 2, 4, 6, 12
“p over q” Find the rational zeros of f(x) = x3 + 2x2 – 11x – 12 p is all of the factors of the constant term. 1, 3, 2, 4, 6, 12 q is all of the factors of the leading coefficient. This one is easy because the leading coefficient is 1 ! The only factor is: 1 = 1, 3, 2, 4, 6, 12

Using Still finding the rational zeros of f(x) = x3 + 2x2 – 11x – 12 1, 3, 2, 4, 6, 12 Possible Zeros: Do synthetic division until you find a zero. x x (-11) x + (–12) 1 k-value 1 • 1 is not a zero to this function. Remainder is not zero so 1 3 -8 1 3 -8 -20

Keep trying Possible Zeros
Test x = -1. x x (-11) x + (–12) -1 • -1 k-value -1 IS a zero to this function! Remainder IS zero so -1 -1 12 1 1 -12 Since -1 is a zero of f, then the result is a factor (x2 + x – 12) This is factorable into: (x + 4)(x – 3). The zeros are: -1 (original zero), -4, 3

The Nightmare Example Find the rational zeros of f(x) = 10x4 - 3x3 - 29x2 + 5x + 12 = 1, 3, 2, 4, 6, 12 1 1, 3, 2, 4, 6, 12 2 1, 3, 2, 4, 6, 12 5 1, 3, 2, 4, 6, 12 10

The Nightmare Example (cont’d)
Finding the zeros of f(x) = 10x4 - 3x3 - 29x2 + 5x + 12 With so many possible zeros, it’s worth our time to get a ballpark answer by graphing the polynomial on the calculator. -3/2 • -15 27 3 -12 10 -18 -2 8 We found the 1st Zero!

= What do you need to remember? Rational Zero Theorem
If f(x) = anxn + … + a1x + a0 (it’s a polynomial) and the polynomial has integer coefficients, then EVERY rational zero of f has the following form: factor of the constant term . factor of the leading coefficient = Rational Zero Theorem Be able to list all possible rational zeros. Then let your calculator do the rest!