Unit 4 Test (Lessons 31–36): Polynomial Functions Solutions

#	Answer	Lesson Origin
1A)	$h (x) = g (x) \cdot f (x) h (x) = (x^{2} - 9) (x - 3) h (x) = x^{3} - 3 x^{2} - 9 x + 27$	31
1B)	The end behavior for a positive third degree polynomial is $x \to - \infty, f (x) \to - \infty, and x \to + \infty, f (x) \to + \infty .$	33
1C)		34
1D)	The function is increasing over the intervals $(- \infty, - 1) and (3, + \infty) .$	35
2)	B	31
3)	C	34
4)	D	34
5)	B	36
6)	A	32
7)	C	31
8)	C	35
9)	B	32
10)	D	33
11)	A	31
12)	A	34, 35
13)	D	36
14)	D	33
15)	B	34
16)	B	31
17)	B	36
18)	C	33
19)	A	32
20)	B	33, 35
21)	A	32
22)	D	31
23)	C	35
24)	B	35
25)	C	36

Answer all parts of the open response problem.

$f (x) = x - 3 g (x) = x^{2} - 9$

The product of g and f form the new function h(x). Find h(x). Show your work.

$h (x) = g (x) \cdot f (x) h (x) = (x^{2} - 9) (x - 3) h (x) = x^{3} - 3 x^{2} - 9 x + 27$

Name the degree and end behavior for h(x).

a > 0, n = 3

The end behavior for a positive third degree polynomial is $x \to - \infty, f (x) \to - \infty, and x \to + \infty, f (x) \to + \infty$ .

Sketch the graph of h(x). Label the roots and turning points.

Note

Remember students should use technology to find the turning points.

Name the interval(s) where h(x) is increasing.

The function is increasing over the intervals $(- \infty, - 1) and (3, + \infty) .$

Multiple Choice

Name the excluded domain values for: $f (x) = \frac{1}{\frac{3}{x} + 1}$

$x \neq - 3$
$x \neq - 3, 0$
$x \neq 0$
$x \neq - 1, 0$

Note

A, C) These options only include one of the two domain restrictions.

The value of the fraction in the denominator equals –1 only when $x = - 3 .$

$1 \div (\frac{3}{x} + 1) = 1 \div (\frac{3}{x} + \frac{x}{x}) 1 \div (\frac{3 + x}{x}), x \neq 0 1 \cdot (\frac{x}{3 + x}) \frac{x}{x + 3}, x \neq - 3, 0$

Name the possible degree n and end behavior.

$n = 3 x \to - \infty, f (x) \to + \infty, and x \to + \infty, f (x) \to + \infty$
$n = 3 x \to - \infty, f (x) \to - \infty, and x \to + \infty, f (x) \to + \infty$
$n = 6 x \to - \infty, f (x) \to + \infty, and x \to + \infty, f (x) \to + \infty$
$n = 6 x \to - \infty, f (x) \to - \infty, and x \to + \infty, f (x) \to + \infty$

bounce = 2, snake = 3, cross through = 1

n = 2 + 3 + 1 = 6

a > 0 from end behavior

Note

A, B) There are 3 roots, but the graph shows multiplicities greater than one with an even end behavior.
B, D) The end behavior is even, but these options are for an odd function.

Determine all possible rational roots for $n (x) = 3 x^{4} - 2 x^{3} + x^{2} - 12 .$

$\pm 1, \pm 3$
$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$
$\pm \frac{1}{3}, \pm \frac{2}{3}, \pm 1, \pm \frac{4}{3}, \pm 2, \pm 4$
$\pm \frac{1}{3}, \pm \frac{2}{3}, \pm 1, \pm \frac{4}{3}, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

$R R T : = \frac{\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12}{\pm 1, \pm 3}$

Note

A–C) These options do not include all possible rational roots.

Select the equation.
y varies directly as x-squared, and inversely as the product of w and z-cubed.

$y = \frac{x^{2}}{w z^{3}}$
$y = \frac{k x^{2}}{w z^{3}}$
$y = \frac{k w x^{2}}{z^{3}}$
$y = \frac{k w z^{3}}{x^{2}}$

$direct variation : y = k x inverse variation : y = \frac{k}{x}$

Note

The constant of variation k is missing.
w belongs in the denominator.
The terms w, x, and z are incorrectly placed in the equation.

Determine the inverse of $p (x) = \frac{1}{\sqrt[3]{x + 6}} .$

$\begin{array}{rcl} p^{- 1} (x) & = & \frac{1}{x^{3}} - 6 \end{array}$
$\begin{array}{rcl} p^{- 1} (x) & = & \frac{1}{x^{3} - 6} \end{array}$
$p^{- 1} (x) = x^{3} - 6$
inverse not shown

Note

The value –6 should not be in the denominator.
The term $x^{3}$ should be in the denominator.

$\begin{array}{rcl} y & = & \frac{1}{\sqrt[3]{x + 6}} \\ x & = & \frac{1}{\sqrt[3]{y + 6}} \\ \frac{(x) (\sqrt[3]{y + 6})}{x} & = & \frac{1}{x} \\ {(\sqrt[3]{y + 6})}^{3} & = & {(\frac{1}{x})}^{3} \\ y + 6 & = & \frac{1}{x^{3}} \\ y & = & \frac{1}{x^{3}} - 6 \end{array}$

Determine the restrictions on the function: $h (x) = \frac{x - 5}{x^{3} - 5 x^{2} - 4 x + 20}$

$x \neq \pm 2$
$x \neq \pm 2, - 5$
$x \neq \pm 2, 5$
$x \neq \pm \sqrt{2}, 5$

$\begin{array}{rcl} (x^{3} - 5 x^{2}) + (- 4 x + 20) & = & 0 \\ x^{2} (x - 5) - 4 (x - 5) & = & 0 \\ (x^{2} - 4) (x - 5) & = & 0 \\ x^{2} & = & 4, x = 5 \\ x & = & \pm 2, 5 \end{array}$

Note

This option is an incomplete list of the restrictions.
The sign is incorrect for 5.
The square root of 4 is ±2, not $\pm \sqrt{2} .$

Name the relative minimum over the interval [0, 3].

(–0.82, –0.68)
(1.47, 0.94)
(0.53, –0.85)
(2, 0)

The relative minimum is the lowest turning point across the given interval

Note

This point is not in the interval.
This point is a relative maximum point.
This is not the lowest turning point over the interval.

Evaluate $[b \circ a] (2) .$

–1
0
2
3

$\begin{array}{rcl} [b \circ a] (2) & = & b (a (2)) \\ = & b (- 1) \\ = & 0 \end{array}$

Note

This option is $a (2) = - 1 .$
This option is $b (1) = 2 .$
This option is the x-value of the intersection point.

The graphs of polynomial functions are ___ smooth, continuous and ___ contain sharp turns or holes.

sometimes, never
always, always
never, sometimes
always, never

By definition, the graphs of a polynomial function are smooth and continuous.

Note

A–C) These options do not correctly define a polynomial function.

Determine (g– h)(–2).

6
4
2
–6

$g (- 2) - h (- 2) 4 - (- 2) 6$

Note

This option is the y-value of the point of intersection.
This option is $[g (h (- 2)] .$
This option is $(h - g) (- 2) .$

Select the graph that contains a multiplicity of two and a multiplicity of three.

cross through = multiplicity 1

bounce = multiplicity 2

snake = multiplicity 3

Note

This graph has a triple root (multiplicity 3) and a single root (multiplicity 1).
This graph has two double roots (multiplicity 2).
This graph has five single roots (multiplicity 1).

p varies jointly as the square of y, and the cube of x.
If $p = - 8, y = 3$ and $x = - 2,$ find p when $y = 1$ and $x = 2 .$

$\frac{1}{9}$
$\frac{4}{9}$
$\frac{2}{9}$
$\frac{8}{9}$

Note

This option is the constant of variation.
This option is the solution when x and y values are switched.
This option is the solution if no exponents are used.

$p = k y^{2} x^{3} - 8 = k {(3)}^{2} {(- 2)}^{3} - 8 = k (9) (- 8) k = \frac{1}{9} p = \frac{1}{9} {(1)}^{2} {(2)}^{3} p = \frac{8}{9}$

Select the graph that does not represent a polynomial function.

A polynomial function is a smooth, continuous graph. An absolute value graph is not a polynomial function.

Note

A–C) These options all represent polynomial functions.

Find the exact roots of $f (x) = x^{4} - 2 x^{3} + x^{2} - 8 x - 12 .$

–3, 1
$\pm 2 i, - 1, 3$
–1, 3
$\pm 2, - 1, 3$

Note

These values have the opposite signs of the roots.
This option is missing a pair of imaginary roots.
This option found the square root of 4 rather than –4.

$RRT : \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

$x^{2} + 4 = 0 x^{2} = - 4 x = \pm 2 i x = \pm 2 i, - 1, 3$

Find the quotient of g and f given: $f (x) = 5 - \frac{1}{x} g (x) = \frac{1}{x}$

$5 x - 1$
$\frac{1}{5 x - 1}$
$\frac{5}{x} - \frac{1}{x^{2}}$
$\frac{2}{x} - 5$

$\frac{g}{f} = \frac{\frac{1}{x}}{5 - \frac{1}{x}} = \frac{1}{x} \div (5 - \frac{1}{x}) (\frac{1}{x}) \div (\frac{5 x}{x} - \frac{1}{x}) = (\frac{1}{x}) \div (\frac{5 x - 1}{x}) (\frac{1}{x}) (\frac{x}{5 x - 1}) = \frac{1}{5 x - 1}$

Note

This option is f divided by g.
This option is fg.
This option is g – f.

Name the end behavior for the polynomial function.
$k (x) = - 3 x {(x - 6)}^{2} (x + 8)$

$x \to - \infty, f (x) \to + \infty, and x \to + \infty, f (x) \to + \infty$
$x \to - \infty, f (x) \to - \infty, and x \to + \infty, f (x) \to - \infty$
$x \to - \infty, f (x) \to - \infty, and x \to + \infty, f (x) \to + \infty$
$x \to - \infty, f (x) \to + \infty, and x \to + \infty, f (x) \to - \infty$

$a < 0, n = 1 + 2 + 1 n = 4$

Note

A, C) Represent functions when $a > 0 .$

Represents a function when n = odd.

The pressure P of gas in a container varies inversely as the volume V. Determine the constant of variation when the pressure is 32 pounds per square inch and the volume is 0.35 cubic feet.

91.428
32.35
11.2
0.001

$P = \frac{k}{V} 32 = \frac{k}{0.35} k = 11.2$

Note

This option is the quotient of P and V.
This option is the sum of P and V.
This option is the quotient of V and P.

Decompose q(x) such that $[f \circ g] (x) = q (x)$ when $q (x) = \frac{x^{2} + 5}{3} .$

$f (x) = \frac{x}{3}, g (x) = x^{2} + 5$
$f (x) = x^{2}, g (x) = \frac{5}{3}$
$f (x) = \frac{x^{2}}{3}, g (x) = 5$
$f (x) = \frac{x}{3}, g (x) = x + 5$

$\begin{array}{rcl} f (x) & = & \frac{x}{3}, g (x) = x^{2} + 5 \\ [f \circ g] (x) & = & [f (g (x))] \\ [f \circ g] (x) & = & \frac{(x^{2} + 5)}{3} \end{array}$

Note

B, C) The result is a constant function.

The x-term is not squared.

Select the sketch of a graph that best represents a fifth degree polynomial function with all real roots.

Graph A has 2 real roots.

Graph C has 3 real roots.

Graph D has 4 real roots.

Note

A, D) These options are even degree functions.

This option is a third degree polynomial.

An item costs x dollars to purchase. What order results in the best price when using a $20 off coupon and a 10% off coupon?
$c (x) = x - 20 d (x) = 0.90 x$

$[c \circ d] (x)$
$[c \circ d] (x) = [d \circ c] (x)$
$[d \circ c] (x)$
cannot be determined

Note

The order of the coupons matters, and they are not equal.
This option does not result in the lowest price.

$If x = 150 \begin{matrix} [c \circ d] (x) = [c (d (x))] & [d \circ c] (x) = [d (c (x))] \\ = [c (d (150))] & = [d (c (150))] \\ = c (135) & = d (130) \\ [c \circ d] (x) = 115 & = 117 \end{matrix}$

If $f (x) = a x^{2} - x - 3$ and $f (- 3) = 18,$ determine f(6).

–81
2
87
63

Note

This option is the solution when $a = - 2 .$
This option is the value of a.
This option occurs if the like terms are not combined correctly.

$\begin{array}{rcl} 18 & = & a {(- 3)}^{2} - (- 3) - 3 \\ 18 & = & 9 a + 3 - 3 \\ 18 & = & 9 a \\ a & = & 2 \\ f (6) & = & 2 {(6)}^{2} - (6) - 3 \\ f (6) & = & 2 (36) - 9 \\ f (6) & = & 63 \end{array}$

For the polynomial function $f (x) = a^{n} x^{n} + . . . + ?,$ name the number of complex roots and the maximum possible number of turning points.

There are n number of complex roots and n turning points.
There are $n - 1$ number of complex roots and n turning points.
There are n number of complex roots and $n - 1$ turning points.
The number of complex roots and turning points cannot be determined.

The Fundamental Theorem of Algebra says that there are n complex roots and $n - 1$ turning points.

Note

This option does not have the correct number of turning points.
This option does not have the correct number of complex roots.

Name all increasing intervals for the given function.

$[- \infty, - 0.82], [- 0.26, 0.53], [1.47, 2]$
$[- 0.82, - 0.26], [0.53, 1.47], [2, + \infty]$
$[- 0.82, - 0.5], [0.53, 1], [2, + \infty]$
$(- 1, 0), (- 0.5, 0), (0, 0), (1, 0), (2, 0)$

Increasing intervals are where the x-values and y-values are both increasing from left to right across the graph.

Note

This option has decreasing intervals.
The intervals stop at the x-intercepts instead of a turning point.
This option includes the x-intercepts, not the intervals.

Determine the description that best matches the equation: $V = \frac{4}{3} π r^{3}$

The volume of a sphere V varies jointly as the cube of the radius r.
The volume of a sphere V varies inversely as the cube of the radius r.
The volume of a sphere V varies directly as the cube of the radius r.
The volume of a sphere V varies jointly as the product of pi and the cube of the radius r.

$direct variation : y = k x k = \frac{4}{3} π$

Note

A, B, D) These options show inverse or joint problems, but this problem is a direct variation problem.