QQuestion
The polynomial 3x³ + 8x² − 15x + k has (x − 1) as a factor. Find the value of k. Hence, factorize the resulting polynomial completely.
✓Answer
Since (x − 1) is a factor, x = 1 makes the polynomial zero.
3(1)³ + 8(1)² − 15(1) + k = 0
⇒ 3 + 8 − 15 + k = 0
⇒ −4 + k = 0
⇒ k = 4
The resulting polynomial is:
3x³ + 8x² − 15x + 4
On dividing (3x3 + 8x2 – 15x + 4) by (x – 1), we get :
3x³ + 8x² − 15x + 4 = (x − 1)(3x² + 11x − 4)
Splitting the middle term:
3x² + 11x − 4 = 3x² + 12x − x − 4
⇒ 3x(x + 4) − 1(x + 4)
⇒ (3x − 1)(x + 4)
Hence:
3x³ + 8x² − 15x + 4 = (x − 1)(3x − 1)(x + 4)
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