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The Rational Root theorem

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Used to factor polynomials when no other method works. The Rational Root theorem. If f(x) is a polynomial with INTEGER coefficients, then the candidates for every rational zero of f has the following form:

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the rational zero theorem
If f(x) is a polynomial with INTEGER coefficients, then the candidates for every rational zero of f has the following form:

Take each factor of p and divide it by each factor of q until all +/- fractional and integer factors can be found.

You will not just divide and get one answer, there will be multiple candidates for zeros

The rational zero theorem
practice using factor theorem
Find all the zeros for:

Take ALL factors of p over ALL factors of q

Practice using Factor Theorem
how can we use the graphing calculator to check zeros
Once you have found some zeros for a function –(use )
  • Plug in a value for x using the store feature
    • Example try x=1, use 1, STO, X (I think of this as saying I want 1 stored as x.)
    • Then you can type in the whole function above and press enter/= and you will get either the remainder or f(x) value.
    • We are looking for zeros, once you try one number you can continue to back your way through the previous entries using 2nd, Enter so you do not have to keep typing in the whole polynomial.
How can we use the graphing calculator to check zeros?
rational root theorem application
Use the rational root theorem to factor the following completely and find all zeros.

Once you get one zero you can use synthetic division to factor and find the other zeros

Ex:

2nd Ex:

Practice pg. 91 #7

Rational root theorem Application
turning points on the graph
Based on the degree of any polynomial, the maximum number of turning points on the graph will be one less than the degree:

Ex: How many turning points does each graph have below?

Turning Points on the graph
homework
Pg. 91 12-13

Pg. 92 2-3, 12-13

OPTIONAL BONUS: pg. 92 #10 (+4)

Homework
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