the rational root theorem the rational root theorem gives us a tool to predict the values of...

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The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots: If P ( x ) a 0 x n a 1 x n 1 ... a n 1 x a n ,where th coeffiecients are all integers, & a rational zero of P( x) in reduced form is p q , the p must be a factor of a n (the constant ter & q must be a factor of a 0 (the leading coef

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Page 1: The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:

The Rational Root Theorem

The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:

If P(x) a0 xn a1 xn 1 ... an 1x an , where the

coeffiecients are all integers,

& a rational zero of P(x) in reduced form is p

q, then

p must be a factor of an (the constant term) &

q must be a factor of a0 (the leading coefficient).

Page 2: The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:

List the Possible Rational Roots

For the polynomial: f (x) x3 3x2 5x 15

All possible values of: p: 1, 3, 5

q: 1

All possible Rational Roots of the form p/q:

p

q: 1, 3, 5

Page 3: The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:

Find a Root That Works

For the polynomial:

Substitute each of our possible rational roots into f(x). If a value, a, is a root, then f(a) = 0. (Roots are solutions to an equation set equal to zero!)

f (1) 1 3 5 15 12

f (3) 27 27 15 15 0 *

f (5) 125 75 25 15 60

f (x) x3 3x2 5x 15

Page 4: The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:

Find the Other Roots

x 3 x 3 3x2 5x 15

Page 5: The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:

Find the Other Roots (con’t)

x 3 x 3 3x2 5x 15x2 5

The resulting polynomial is a quadratic, but it doesn’t have real factors. Solve the quadratic set equal to zero by either using the quadratic formula, or by isolating the x and taking the square root of both sides.

Page 6: The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:

Find the Other Roots (con’t)

The solutions to the quadratic equation: x i 5, i 5

The three complex roots of the polynomial are: x 3, i 5, i 5

For the polynomial: f (x) x3 3x2 5x 15

Page 7: The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:

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