Summary
- To factor a polynomial, \(P(x)\), of degree \(3\) or greater, begin with the factor theorem. Look for a value \(n\), such that \(P(n)=0\).
- To employ the rational root theorem (also called the rational zero theorem), the test values for \(n\) must be of the form \(\frac{q}{r}\), where \(q\) is a factor of the constant term of \(P(x)\) and \(r\) is a factor of the leading coefficient. If \(P\left(\frac{q}{r}\right)=0\), then \(rx-q\) is a factor of \(P(x)\).
- To determine the other factor(s) of \(P(x)\), divide \(rx-q\) into the polynomial using long division (or synthetic division) or use the “have and need” method.
- This process may need to be repeated several times depending on the degree of the polynomial.
- It is possible that no linear factors will be found depending on the nature of the polynomial.
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