Finding Rational Roots of Polynomials
After learning about the Factor Theorem, we know that finding a factor is the same as finding a zero (root) of the polynomial . But how do we find the value of , especially if the polynomial has a high degree?
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After learning about the Factor Theorem, we know that finding a factor is the same as finding a zero (root) of the polynomial . But how do we find the value of , especially if the polynomial has a high degree?
Trying out all numbers is certainly not efficient. This is where the Rational Zero Theorem (or Rational Root Theorem) comes into play. This theorem helps us narrow down the list of possible rational roots of a polynomial.
Let be a polynomial where all coefficients () are integers, with and .
If the polynomial has a rational zero (root) of the form (where and are integers, , and is a fraction in simplest form), then:
This theorem only provides a list of possible rational roots. Not all values of from the list are necessarily actual roots of the polynomial. We still need to test them.
Here are the steps to find rational roots using this theorem, often combined with the Factor Theorem:
Factor the polynomial completely.
Identify Coefficients:
The coefficients are integers. and .
Factors of (from ):
Factors of (from ):
Possible Roots :
Dividing all by yields:
Test Possible Roots: Let's test some values from the list.
Try :
Factor Further (using the root ):
Divide by using Horner's ().
Factor the Quotient:
Factor .
Complete Factorization:
Factor completely using the Rational Zero Theorem and the Factor Theorem.
Identify Coefficients: , .
Factors of (from ): .
Factors of (from ): .
Possible Roots : .
Test Roots:
Try .
Divide using Horner ():
Factor the Quotient:
Factor .
Complete Factorization:
Try :
Try :
Try :
Success! So, is a root, and is a factor.
Alternatively, try :
Success! So, is a root, and is a factor.
The quotient is .
Thus, .
Since , is a root and is a factor.
The quotient is .
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