Precalculus Polynomial Division and the Factor Theorem Test Solutions and Scoring (PDF)

Quotient \(x^{2}+x-12=(x+4)(x-3)\), remainder 0.

\((x-2)(x+4)(x-3)\)

Scoring: 1 point for the division, 1 point for factoring the quotient.

Coefficients \(1,2,0,-1,4\) with \(c=-1\): bottom row \(1,1,-1,0,4\).

\(x^{3}+x^{2}-x+\frac{4}{x+1}\)

Scoring: 1 point for the bottom row (including the 0 placeholder), 1 point for the written result.

\(f(3)=54-45+3-7=5\).

5

Scoring: 1 point for substituting 3, 1 point for the value.

\(f(-1)=-1+k+2+8=k+9=0\), so \(k=-9\).

\(k=-9\)

Scoring: 1 point for setting f(-1) = 0, 1 point for k.

(a) \(\pm1,\pm2,\pm3,\pm6,\pm\frac12,\pm\frac32\). (b) Bottom row \(2,5,-3,0\): remainder 0. (c) \(2x^{2}+5x-3=(2x-1)(x+3)\).

\(x=2,\ \frac12,\ -3\)

Scoring: 1 point for each part.

Synthetic division with \(c=2\): \(4,5\) and remainder \(11\).

\(4x^{2}-3x+1=(x-2)(4x+5)+11\)

Scoring: 1 point for q(x), 1 point for r.

\(u=x^{2}\): \(u^{2}-13u+36=(u-4)(u-9)\).

\(x=\pm2,\ \pm3\)

Scoring: 1 point for the substitution and factoring, 1 point for all four zeros.

(a) Synthetic division with \(c=-2\): \(1,12,44,48\to 1,10,24\), remainder 0, quotient \(x^{2}+10x+24\). (b) \((x+4)(x+6)\). (c) \(5\cdot 7\cdot 9=315\).

\(x^{2}+10x+24\); \(x+4\) and \(x+6\); \(315\text{ in}^{3}\)

Scoring: 1 point for each part.

\((2x+1)(x-1)(x-4)=(2x+1)(x^{2}-5x+4)=2x^{3}-9x^{2}+3x+4\).

\(2x^{3}-9x^{2}+3x+4\)

Scoring: 1 point for the factors (with 2x + 1), 1 point for the expansion.