Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
The last six problems use crates, kit costs, a thrown ball and a medicine dose, so every inequality and asymptote has a real meaning.
Standards: A-APR.B.2, A-APR.B.3, F-IF.C.7c, F-IF.C.7d, N-CN.C.9 · Level: Grade 12 Precalculus
Problem 1 List Possible Rational Zeros ★★★
List every possible rational zero of P(x) = 3x3 − x2 + 5x − 10.
Problem 2 End Behavior from the Leading Term ★★★
Describe the end behavior of each function in words. (a) f(x) = −2x5 + 7x − 1 (b) g(x) = 4x4 − x3 (c) h(x) = −x6 + 9x2
Problem 3 Conjugate Zeros ★★★
Suppose P has degree 4 and real coefficients, and two of its zeros are 2 − 5i and −1 + i. Name the remaining two zeros.
Problem 4 Classify the Asymptote Type ★★★
Without graphing, state whether each function has a horizontal asymptote, a slant asymptote or neither, and give its equation. (a) x3 + 1x2 − 4 (b) 3x2 + 1x2 − 9 (c) x + 5x2 + 2
Problem 5 Read a Sign Chart ★★★
The sign chart shows the sign of Q(x) = (x + 5)(x − 1)(x − 4) on each interval. Solve Q(x) > 0.
Problem 6 Count the Complex Zeros ★★★
Let P(x) = (x − 4)3(x2 + 9)(x + 1). Give the degree, list all complex zeros counting repeats, and confirm the Fundamental Theorem of Algebra count.
Problem 7 Find All Zeros with Candidates ★★★
Find all zeros of P(x) = 2x3 + 7x2 − 5x − 4.
Problem 8 Real and Complex Zeros Together ★★★
Determine every zero, real and non-real, of P(x) = x3 + x2 + 4x + 30.
Problem 9 Build a Polynomial from Zeros ★★★
Construct a monic cubic with real coefficients whose zeros include 5 and 3 − 2i. Multiply out the factors.
Problem 10 Slant Asymptote by Division ★★★
Find the slant asymptote and the vertical asymptote of f(x) = 3x2 − x + 4x − 2.
Problem 11 Identify Asymptotes on a Graph ★★★
The graph shows f(x) = x2 − 4x + 1x − 3 with two dashed asymptotes. Write their equations, then confirm the slant one by division.
Problem 12 Quadratic Inequality ★★★
The graph of y = 2x2 − 5x − 12 is shown. Solve 2x2 − 5x − 12 ≤ 0 algebraically and give the solution set as an interval.
Problem 13 Rational Inequality with a Constant ★★★
Solve x + 1x − 5 ≤ 2.
Problem 14 Zeros and Multiplicity from a Graph ★★★
The graph is a polynomial with leading coefficient −1 and degree 4. It meets the x-axis at −2, 1 and 3 and touches (does not cross) at 1. Find each zero’s multiplicity, describe the end behavior, and write a possible equation.
Problem 15 Solve a Quartic over the Complex Numbers ★★★
Solve x4 − x2 − 12 = 0 over the complex numbers.
Problem 16 Dimensions of a Crate ★★★
A crate has width x feet, length x + 2 feet and height x − 1 feet. Its volume is 72 cubic feet. Set up the equation, find the real value of x with the Rational Root Theorem, and show that the other two solutions are not real.
Problem 17 Average Cost and Its Asymptote ★★★
A robotics club builds n kits for a total cost of 2n2 + 30n + 450 dollars. The average cost per kit is A(n) = 2n2 + 30n + 450n. Find the slant asymptote, then find how many kits (whole numbers) keep the average cost at or below $110.
Problem 18 Ball Above a Height ★★★
A ball is thrown so that its height is h(t) = −16t2 + 48t + 64 feet after t seconds. For how long is it above 32 feet before it lands? Give the exact interval and a decimal approximation.
Problem 19 Medicine Concentration Window ★★★
The concentration of a medicine in the blood is C(t) = 40tt2 + 16 mg/L after t hours. When is the concentration at least 2.5 mg/L? Give exact endpoints and approximations.
Problem 20 Use a Given Zero to Solve a Quartic ★★★
Given that 2i is a zero of P(x) = x4 − 2x3 + 6x2 − 8x + 8, find every zero.
