Precalculus Polynomial Functions and Their Graphs Test Solutions and Scoring (PDF)

Degree 3 with leading coefficient \(-5\): odd degree and negative, so the left end rises and the right end falls.

Degree 3; \(-5\); \(f(x)\to\infty\) as \(x\to-\infty\), \(f(x)\to-\infty\) as \(x\to\infty\)

Scoring: 1 point for degree and leading coefficient, 1 point for end behavior.

\(-2\): multiplicity 2, touches. \(5\): multiplicity 1, crosses.

\(-2\) (touch), \(5\) (cross)

Scoring: 1 point for each zero.

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

\(x=4,\ 3,\ -3\)

Scoring: 1 point for factoring by grouping, 1 point for all three zeros.

\((x+2)(x-1)(x-3)=(x^{2}+x-2)(x-3)=x^{3}-2x^{2}-5x+6\).

\(x^{3}-2x^{2}-5x+6\)

Scoring: 1 point for the factors, 1 point for the expansion.

(a) \(-2,\ 1,\ 3\). (b) Three simple zeros give degree 3; the left end rises and the right end falls, so \(a< 0\). (c) \(g(x)=a(x+2)(x-1)(x-3)\) and \(g(0)=6a=-6\), so \(a=-1\).

\(-2,1,3\); degree 3, \(a< 0\); \(g(x)=-(x+2)(x-1)(x-3)\)

Scoring: 1 point for each part.

Zeros \(-1,3,5\). Test: \(x=-2\) gives negative; \(x=0\) gives \((1)(-3)(-5)>0\); \(x=4\) gives \((5)(1)(-1)< 0\); \(x=6\) gives positive. Include the zeros.

\((-\infty,-1]\cup[3,5]\)

Scoring: 1 point for the sign test, 1 point for correct intervals with endpoints.

\(f(1)=1+2-5=-2\) and \(f(2)=8+4-5=7\). The values have opposite signs and \(f\) is continuous.

A zero lies between 1 and 2

Scoring: 1 point for each correct value, shown with the sign conclusion in the second point.

(a) \(V(1)=1(8)(14)=112\); \(V(2)=2(6)(12)=144\). (b) \(0< x< 5\). (c) \(V(3)=3(4)(10)=120\); the greatest is at \(x=2\).

\(112,\ 144\); \(0< x< 5\); \(120\); \(x=2\)

Scoring: 1 point for each part.

Degree \(2+3+1=6\). \(x=0\): even, touches. \(x=3\): odd, crosses (flattens). \(x=-1\): odd, crosses.

Degree 6; touch at 0, cross at 3 and \(-1\)

Scoring: 1 point for the degree, 1 point for the behavior at the three zeros.