Exam 2: Polynomial, Power, and Rational Functions

Solve the problem. -The polynomial function $\mathrm { I } ( \mathrm { t } ) = - 0.1 \mathrm { t } ^ { 2 } + 1.9 \mathrm { t }$ represents the yearly income (or loss) from a real estate investment, where $t$ is time in years. After how many years does income begin to decline? Round to the nearest tenth of a year, if necessary.

Use synthetic division to determine whether the number k is an upper or lower bound (as specified) for the real zeros of the function f. - $\mathrm { k } = 3 ; \mathrm { f } ( \mathrm { x } ) = \mathrm { x } ^ { 4 } + 4 \mathrm { x } ^ { 3 } - 3 \mathrm { x } ^ { 2 } - 4 \mathrm { x } + 5$ ; Upper bound?

Write the word or phrase that best completes each statement or answers the question. Write a sentence that expresses the relationship in the formula, using the language of variation or proportion. -P = nb, where P is the perimeter of a regular polygon with n sides each of length b.

Provide an appropriate response. -Sketch the graph of a polynomial function which has a negative $y$ -intercept, one local maximum and one local minimum, but has no absolute extrema.

Find all of the real zeros of the function. Give exact values whenever possible. Identify each zero as rational or irrational. - $f ( x ) = x ^ { 4 } + 2 x ^ { 3 } - 17 x ^ { 2 } - 4 x + 30$

Divide using synthetic division, and write a summary statement in fraction form. - $\frac { 2 x ^ { 5 } - x ^ { 4 } + 3 x ^ { 2 } - x + 5 } { x - 1 }$

Write the word or phrase that best completes each statement or answers the question. Graph the rational function and analyze it in the following way: find the intercepts, asymptotes, use limits to describe the behavior at the vertical asymptotes and the end behavior. Find the domain and range. Determine where the function is continuous and where it is increasing and decreasing. Find any local extrema. - $f ( x ) = \frac { x ^ { 2 } - x - 6 } { x - 2 }$

Write the function as a product of linear and irreducible quadratic factors, all with real coefficients. - $f ( x ) = 3 x ^ { 3 } - 20 x ^ { 2 } - 6 x - 7$

Write the function as a product of linear and irreducible quadratic factors, all with real coefficients. - $f ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } + 7 x + 12$

State how many complex and real zeros the function has. - $f ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 2 x + 1$

Divide using synthetic division, and write a summary statement in fraction form. - $\frac { 2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 3 x } { x + 3 }$

Describe how to transform the graph of an appropriate monomial function f(x) = xn into the graph of the given polynomial function. Then sketch the transformed graph. - $g ( x ) = 5 ( x - 3 ) ^ { 4 } - 2$

List the x- and y-intercepts, and graph the function. - $f ( x ) = \frac { - 3 } { x - 6 }$

Determine the x values that cause the polynomial function to be (a) zero, (b) positive, and (c) negative. - $f ( x ) = ( x + 4 ) ^ { 3 } \left( 4 x ^ { 2 } + 5 \right) ( x - 4 ) ^ { 4 }$

Divide f(x) by d(x), and write a summary statement in the form indicated. - $f ( x ) = x ^ { 4 } + 4 x ^ { 3 } + 7 x ^ { 2 } + 32 x + 24 ; d ( x ) = x ^ { 2 } + 4 x + 3 \quad$ (Write answer in polynomial form)

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. - $- 4 i$ and $\sqrt { 3 }$

Evaluate the limit based on the graph of f shown. - $\lim _ { x - 5 ^ { + } } f ( x )$

Solve the problem. -Find the remainder when $x ^ { 56 } - 20$ is divided by $x - 1$ .

Solve the problem. -The first column shows the independent variable $( x )$ . The second column shows the dependent variable (y). Find the quadratic regression model.

State the power and constant of variation for the function, and then analyze it. - $f ( x ) = 4 x ^ { - 3 }$