QQuestion
Factorize completely using the Factor Theorem: 2x³ − x² − 13x − 6.
✓Answer
Let f(x) = 2x³ − x² − 13x − 6.
Substituting x = −2:
⇒ f(−2) = 2(−2)³ − (−2)² − 13(−2) − 6
⇒ f(−2) = 2(−8) − 4 + 26 − 6
⇒ f(−2) = −16 − 4 + 26 − 6
⇒ f(−2) = 0
Therefore, x + 2 is a factor.
Dividing, 2x3 – x2 – 13x – 6 by x + 2, we get :
Hence:
⇒ 2x³ − x² − 13x − 6 = (x + 2)(2x² − 5x − 3)
Splitting the middle term:
⇒ 2x² − 5x − 3 = 2x² − 6x + x − 3
⇒ 2x(x − 3) + 1(x − 3)
⇒ (2x + 1)(x − 3)
Therefore, 2x³ − x² − 13x − 6 = (x + 2)(2x + 1)(x − 3).
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