SAT Math — polynomial functions: factors, zeros and end behavior

The College Board lists polynomial equations and functions in the Advanced Math domain of the digital SAT, and the harder questions go beyond quadratics: a cubic to factor, a constant to find from a known factor, a remainder to state without long division, or a graph to describe from its factors. This material trains those skills.

You begin with factoring a cubic by grouping and listing all of its zeros, then use the factor theorem to find a missing coefficient when x − 2 is known to be a factor. The remainder theorem appears twice: once to compute the remainder of a division by evaluating the polynomial, and once to decide what must be true when a division leaves a remainder of 7. A question gives four values of a cubic and asks which binomial must be a factor.

Graph questions are answered from the algebra. You decide how a graph behaves at a zero of multiplicity two, describe the end behavior of a cubic with a negative leading coefficient, count the x-intercepts of a quartic written as a quadratic in x², and identify which statement about a factored cubic is true. You also write a polynomial from its zeros, solve x³ = 9x without losing the solution x = 0, and evaluate a box-volume polynomial in context, where the three factors are the height, the width and the length of an open box folded from a sheet of cardboard.

The material offers a quiz of twelve four-option questions, each with an explanation that shows the method and names the slip behind each wrong option, and a set of flashcards that collects the rules to remember.

The content is based on the College Board's published skill descriptions for the Advanced Math domain of the digital SAT Math section. It is independent practice and is not produced or endorsed by the College Board.

• Factor cubic polynomials by grouping and list all zeros
• Use the factor theorem to find an unknown coefficient
• Use the remainder theorem to find or interpret a remainder
• Relate multiplicity to whether a graph touches or crosses the x-axis
• Describe end behavior from the leading term and write a polynomial from its zeros

Practice material written by Zestly, based on the College Board's published description of the digital SAT Math section (Advanced Math domain: polynomial equations and nonlinear functions), as given on satsuite.collegeboard.org and in the Assessment Framework for the Digital SAT Suite.