JAMB 2000 · UME · Q19

Polynomials & algebraic division

If $(x - 1)$ , $(x + 1)$ and $(x - 2)$ are factors of the polynomial $ax^3 + bx^2 + cx - 1$ , find $a$ , $b$ , $c$ respectively.

Worked solution (try it first)

By the factor theorem, $f(1) = 0$ : $a + b + c - 1 = 0$ .
And $f(-1) = 0$ : $-a + b - c - 1 = 0$ .
Add these: $2b - 2 = 0$ , so $b = 1$ and then $a + c = 0$ , so $c = -a$ .
$f(2) = 0$ : $8a + 4b + 2c - 1 = 0$ .
Put in $b = 1$ and $c = -a$ : $6a + 3 = 0$ , so $a = -\frac12$ .
Then $c = \frac12$ , so $a, b, c$ are $-\frac12, 1, \frac12$ , option A.

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