Polynomial and Rational Functions in Depth: Precalculus Math Test – PDF Download

Question 1: Rational Root Theorem (3 points)

Let \(P(x)=2x^{3}+x^{2}-13x+6\).

1. How many possible rational zeros does the Rational Root Theorem give? (1 point)
2. Find all zeros of \(P\). (2 points)

Question 2: Complex zeros of a quartic (2 points)

Find all real and complex zeros of \(f(x)=x^{4}-16\).

Question 3: Build a polynomial (2 points)

Write a monic cubic polynomial with real coefficients that has zeros \(4\) and \(1+3i\). Give the answer in standard form.

Question 4: Read a polynomial graph (2 points)

The graph shows a polynomial \(g\) with a zero at \(-2\) and a zero at \(1\).

1. State the multiplicity of each zero as odd or even. (1 point)
2. State the least possible degree and the sign of the leading coefficient. (1 point)

Question 5: Asymptotes from a graph and a rule (2 points)

The graph shows \(f(x)=\frac{x^{2}-2x-3}{x+2}\).

1. Find the equation of the vertical asymptote. (1 point)
2. Find the equation of the slant asymptote. (1 point)

Question 6: Cubic inequality (3 points)

Solve \((x+3)(x-1)(x-4) > 0\) and write the answer in interval notation.

Question 7: Rational inequality (3 points)

Solve \(\frac{x-5}{x+1} \le 2\) and write the answer in interval notation.

Question 8: Average cost model (2 points)

The average cost per shirt for a team store is \(A(x)=\frac{15x+900}{x}\) dollars for \(x\) shirts.

1. Find the horizontal asymptote of \(A\) and interpret it. (1 point)
2. Find the smallest number of shirts for which the average cost is at most $20. (1 point)

Question 9: Conjugate zeros (1 point)

A polynomial with real coefficients has degree 5 and zeros \(2\), \(1-i\) and \(3i\). Write the other two zeros.

See every chapter on the Precalculus math page.