MCQ
The factors of$ x^3 - 7x + 6$ are:
- A$x(x - 6)(x - 1)$
- B$(x^2 - 6)(x - 1)$
- C$(x + 1)(x + 2)(x + 3)$
- ✓$(x - 1)(x + 3)(x - 2)$
✓
Answer
Correct option: D.
$(x - 1)(x + 3)(x - 2)$
$x^3 - 7x + 6 = x^3 - 7x + 6 + 1 - 1 ($by adding $+1 -1$ to $\text{R.H.S})$
$= x^3 - 7x + 7 - 1$
$= (x^3 - 1) - 7(x - 1)$
Now by identity $a^3 - b^3 = (a - b)(a^2 + b^2 + ab),$ we get
$x^3 - 7x + 6 = (x^3 - 1) - 7(x - 1)$
$= (x - 1)(x^2 + x + 1) - 7(x - 1)$
$= (x - 1)(x^2 + x + 1 - 7)$
$= (x - 1)(x^2 + x - 6)$
$= (x - 1)(x + 3)(x - 2)$
$= x^3 - 7x + 7 - 1$
$= (x^3 - 1) - 7(x - 1)$
Now by identity $a^3 - b^3 = (a - b)(a^2 + b^2 + ab),$ we get
$x^3 - 7x + 6 = (x^3 - 1) - 7(x - 1)$
$= (x - 1)(x^2 + x + 1) - 7(x - 1)$
$= (x - 1)(x^2 + x + 1 - 7)$
$= (x - 1)(x^2 + x - 6)$
$= (x - 1)(x + 3)(x - 2)$
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