Rational Root Theorem Math 3 MM3A1.

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Presentation transcript:

Rational Root Theorem Math 3 MM3A1

To find the zeroes of a Polynomial: The Rational Root Theorem: If f(x) = anxn + . . . + a1x + a0 has integer coefficients, then every rational zero of f(x) has the following form: p = factor of constant term a0 q factor of leading coefficient an

Use the rational root theorem to determine all possible rational roots: What is the constant? -10 This is going to be the numerator What is the leading coefficient? 2 This is going to be the denominator Possible roots:

Things to consider: The degree of the polynomial will tell you the max number of zeroes Some roots may be imaginary Use factors of the constant term (a0) as your numerator Use factors of the leading coefficient (an) as your denominator Use your calculator and synthetic division to determine if the possible ratios are actual zeroes

Find the rational zeroes of: Possible: Use your calculator to graph the function and determine which possible zeroes to try. Which ones look like they may work? Use synthetic division to check them

Actual Solutions 2 -3/2 1/4 Try 2 2 8 -6 -23 6 2 8 -6 -23 6 16 20 -6 8 10 -3 0 now use, the smaller polynomial: Try -3/2 -3/2 8 10 -3 -12 3 8 -2 0 2 -3/2 1/4 Actual Solutions

Now, you can give the factored form of the equation The zeroes for are { 2, -3/2, ¼ } So, the factored form of the equation is:

Find the rational zeroes of: Possible: Use your calculator to graph the function and determine which possible zeroes to try. Which ones look like they may work? Use synthetic division to check them

Actual Solutions 3 -3/2 1/3 Try 3 3 6 -11 -24 9 3 6 -11 -24 9 18 21 -9 6 7 -3 0 now use, the smaller polynomial: Try -3/2 -3/2 6 7 -3 -9 3 6 -2 0 3 -3/2 1/3 Actual Solutions

Now, you can give the factored form of the equation The zeroes for are { 3, -3/2, 1/3 } So, the factored form of the equation is: