Polynomial and Rational Functions Review: math practice, Grade 12 – download the PDF

1 Degree and end behavior ★★★

Let \(f(x)=-2x^5+4x^2-7\). Give its degree, leading coefficient, and \(y\)-intercept, then describe its end behavior.

2 Four end behaviors ★★★

Describe the end behavior of each function: (a) \(f(x)=4x^3-x+1\); (b) \(g(x)=-x^4+3x^2\); (c) \(h(x)=-6x^3+x^2\); (d) \(k(x)=x^6-2x\).

3 Zeros and multiplicities ★★★

Find the zeros of \(f(x)=x^2(x-4)(x+1)^3\), give each multiplicity, and say whether the graph crosses or touches the \(x\)-axis there. What is the degree of \(f\)?

4 A first synthetic division ★★★

Use synthetic division to divide \(x^3-4x^2+x+6\) by \(x-2\). State the quotient and the remainder, then write the polynomial as a product.

5 Remainders by evaluating ★★★

Let \(P(x)=x^3-5x+4\). Without dividing, find the remainder when \(P(x)\) is divided by (a) \(x-3\) and (b) \(x+1\).

6 Simple asymptotes ★★★

Find the vertical and horizontal asymptotes of \(f(x)=\dfrac{3x}{x+4}\).

7 A quick inequality ★★★

Solve \((x-5)(x+2)\lt0\) and write the answer in interval notation.

8 True or false? ★★★

Decide whether each statement is true or false, and justify it. (a) A degree-4 polynomial with real coefficients can have no real zeros. (b) A degree-3 polynomial with real coefficients always has at least one real zero. (c) \(f(x)=\dfrac{x^2-1}{x-1}\) has a vertical asymptote at \(x=1\).

9 Factor Theorem ★★★

Show that \(x+3\) is a factor of \(P(x)=x^3+2x^2-5x-6\), then factor \(P(x)\) completely.

10 Finding a missing coefficient ★★★

The polynomial \(P(x)=x^3+kx^2-3x-10\) has \(x-2\) as a factor. Find \(k\), then find all the zeros of \(P\), real and complex.

11 Rational Root Theorem at work ★★★

List all possible rational zeros of \(2x^3+x^2-13x+6\), then find all of its zeros.

12 An open-top box ★★★

A square sheet of cardboard measures 12 inches (about 30.5 cm) on each side. Squares of side \(x\) inches are cut from the four corners, and the sides are folded up to make an open box. (a) Show that the volume is \(V(x)=x(12-2x)^2\). (b) Find the value of \(x\) that gives a volume of 128 cubic inches.

13 Hole or asymptote? ★★★

Let \(f(x)=\dfrac{2x^2-8}{x^2-x-6}\). Factor, then find the hole, the vertical asymptote, the horizontal asymptote, and both intercepts.

14 Sign chart for a quotient ★★★

Solve \(\dfrac{x+4}{x-2}\ge0\).

15 Slant asymptote ★★★

Find the slant asymptote and the vertical asymptote of \(f(x)=\dfrac{2x^2+3x-5}{x+2}\).

16 Build a polynomial ★★★

Write a polynomial \(P\) with real coefficients, with zeros \(-3\), \(1\) (multiplicity 2), and \(2-i\), of smallest possible degree, and with \(P(0)=30\). Give its degree and end behavior.

17 From a graph to an equation ★★★

The graph of a cubic polynomial \(f\) is shown.

(a) Use the zeros and their multiplicities to write \(f(x)=a(\dots)\). (b) Use the \(y\)-intercept to find \(a\). (c) Expand \(f(x)\).

18 Zero given, find the rest ★★★

The polynomial \(P(x)=x^4+x^3-5x^2+x-6\) has the zero \(i\). Find all its zeros.

19 A rational inequality with a twist ★★★

Solve \(\dfrac{x+1}{x-2}\ge2\). Explain why you should not multiply both sides by \(x-2\).

20 A quartic inequality ★★★

Solve \(x^4-5x^2+4\le0\).

21 Average cost ★★★

A workshop has a fixed cost of 1,200 dollars and spends 8 dollars on materials for each item. The average cost per item for \(x\) items is \(A(x)=\dfrac{8x+1200}{x}\), \(x\gt0\). (a) Find the horizontal asymptote and interpret it. (b) How many items must be made so that the average cost is at most 10 dollars?

22 Missing term, fractional divisor ★★★

Let \(P(x)=2x^3-5x+1\). Use synthetic division with \(c=\tfrac12\) to divide \(P(x)\) by \(x-\tfrac12\). What is \(P\bigl(\tfrac12\bigr)\)? Is \(x-\tfrac12\) a factor?