If \((x – 5)\) is a factor of \(x^3 – 4x^2 – 11x + 30\), find the remaining factors.
-
\((x + 3) and (x - 2)\)
- \((x - 3) and (x + 2)\)
- \((x - 3) and (x - 2)\)
- \((x + 3) and (x + 2)\)
The correct answer is: A
Explanation
(x - 5) is a factor of \(x^3 - 4x^2 - 11x + 30\). To find the remaining factors, let's draw out \((x - 5)\) from the parent expression.
\(x^3 - 4x^2 - 11x + 30 = x^3 - 5x^2 + x^2 - 5x - 6x + 30\)
\(= x^2(x - 5) + x(x - 5) - 6(x - 5) = (x - 5)(x^2 + x - 6)\)
∴ To find the remaining factors, we factorize \((x2 + x - 6)\)
\(x^2 + x - 6 = x^2 + 3x - 2x - 6\)
\(= x(x + 3) - 2(x + 3) = (x + 3)(x - 2)\)
∴ The other two factors are \((x + 3) and (x - 2)\)
ALTERNATIVELY
\(∴ x^2 + x - 6 = (x + 3) and (x - 2)\)