WAEC 2014 · Paper 2 · Q9

(a)
Differentiate $(x - 3)(x^2 + 5)$ with respect to $x$ .
(b)(i)
If $(x + 1)^2$ is a factor of $f(x) = x^3 + ax^2 + bx + 3$ , where $a$ and $b$ are constants, find the values of $a$ and $b$ .
Separate values with commas, e.g. 3, −2
(b)(ii)
Find the zeros of $f(x)$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

Expand first: $(x - 3)(x^2 + 5) = x^3 - 3x^2 + 5x - 15$ .
Differentiate term by term: $\dfrac{dy}{dx} = 3x^2 - 6x + 5$ .

(b)(i)

Divide $x^3 + ax^2 + bx + 3$ by $(x + 1)^2 = x^2 + 2x + 1$ : the quotient is $x + (a - 2)$ .
The remainder is $(b - 2a + 3)x + (5 - a)$ .
$(x + 1)^2$ is a factor, so the remainder is zero: $5 - a = 0$ and $b - 2a + 3 = 0$ .
So $a = 5$ , and then $b = 2(5) - 3 = 7$ .

(ii)

$f(x) = x^3 + 5x^2 + 7x + 3$
$= (x + 1)^2(x + 3)$ , since the quotient is $x + (5 - 2) = x + 3$ .
So the zeros of $f(x)$ are $-1$ (twice) and $-3$ .

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