ACT Preparation / Unit 11 ACT 1025 worked example, 60 to 90 seconds
Using the Factor Theorem
The idea
Factor-theorem questions are answered in three steps. First, identify c from the proposed factor: x − c gives c, and x + c gives −c. Second, evaluate the polynomial at c, keeping any unknown constant as a letter. Third, set the value equal to zero to make it a factor, or read it as the remainder, and solve for the constant if there is one.
In practice
Worked example: for what k is x + 2 a factor of x³ + kx² − x + 2? Step one: c = −2. Step two: (−2)³ + k(−2)² − (−2) + 2 = −8 + 4k + 2 + 2 = 4k − 4. Step three: 4k − 4 = 0, so k = 1. Check: x³ + x² − x + 2 at x = −2 is −8 + 4 + 2 + 2 = 0.
Where people slip
The sign of c is the slip to watch: a factor of x + 2 means evaluating at −2, not at 2.
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