Use a known zero to determine a coefficient and reduce a cubic.
Notes and key methods
The factor theorem says x−a is a factor of P if and only if \(\displaystyle P(a)=0. An\) unknown coefficient can be found from this condition before any division.
Divide out the known factor and factor or solve the remaining quadratic. The remainder theorem is the related statement that the remainder on division by x−b is P(b).
Check the conditions
Use the zero of the factor, with its correct sign.
Check all recovered roots in the original equation.
Three worked examples
Example 1 · Factor theorem and cubic equations
\(\displaystyle P(x)=\)\(k x + x^{3} + 48\). It is given that \(x - 2\) is a factor. Let \(\displaystyle u=2\).
Find k using the given factor.
Factor P completely and solve \(\displaystyle P(x)=0\).
Find the negative real root of P.
Find the remainder on division by x−u.
Show the reasoning
(a)
\(2 k + 56 = 0\) determines k uniquely since \(\displaystyle a\ne 0\).
Answer: \(-28\)
(b)
Divide out x−a, factor the quadratic and set every factor equal to zero.