Math · College algebra · Worked example
Use the remainder and factor theorems
Find the remainder when P(x) = x⁴ − 3x³ + 2x − 5 is divided by x − 2. Then decide whether x + 1 is a factor of x³ + 4x² + x − 2, and find that polynomial’s zeros.
Evaluate instead of dividing
By the remainder theorem, the remainder on dividing by x − 2 is P(2).
Test the factor
x + 1 = x − (−1), so evaluate the second polynomial at −1. A value of 0 means x + 1 is a factor.
Divide out the factor
Synthetic division by x + 1, with c = −1, on the coefficients 1, 4, 1 and −2 gives the sums 1, 3, −2 and 0.
Finish the zeros
x² + 3x − 2 does not factor over the integers; the quadratic formula gives x = (−3 ± √17)/2, about 0.56 and −3.56. The zeros are −1 and (−3 ± √17)/2.
Result
The remainder is P(2) = −9. x + 1 is a factor: x³ + 4x² + x − 2 = (x + 1)(x² + 3x − 2), with zeros −1 and (−3 ± √17)/2.
Your turn
Is x − 2 a factor of x³ − 4x² + x + 6? If it is, factor the polynomial completely.
Show the answer and explanation
Yes: x³ − 4x² + x + 6 = (x − 2)(x − 3)(x + 1).
The value at 2 is 8 − 16 + 2 + 6 = 0, so x − 2 is a factor. Dividing it out leaves x² − 2x − 3 = (x − 3)(x + 1).
Keep exploring
Graph plots x³ + 4x² + x − 2 with its three zeros marked.
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