WAEC 2008 · Paper 2 · Q3

Polynomials & quadratic roots

(a)
The remainder when the polynomial $px^4 + qx^3 - 8x^2 + 6$ is divided by $(x^2 - 1)$ is $(2x + 1)$ . Find the values of the constants $p$ and $q$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

Write $px^4 + qx^3 - 8x^2 + 6 = (x^2 - 1)Q(x) + 2x + 1$ .
At $x = \pm1$ , $x^2 - 1 = 0$ , so only the remainder is left.
Put $x = 1$ : $p + q - 8 + 6 = 2 + 1$ , so $p + q = 5$ .
Put $x = -1$ : $p - q - 8 + 6 = -2 + 1$ , so $p - q = 1$ .
Add the equations: $2p = 6$ , so $p = 3$ .
Then $q = 5 - 3 = 2$ .

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