PSAT Advanced Math — polynomial arithmetic, factoring and zeros

"Equivalent expressions" is one of the three skill areas of the Advanced Math domain on the PSAT/NMSQT. At this level the College Board describes it as fluent addition, subtraction and multiplication of polynomials and factoring limited to three moves: taking out a common factor, rewriting a difference of two squares, and rewriting a trinomial as the product of two binomials. This material trains exactly those moves, and then connects the factored form of a polynomial function with its zeros and its graph.

On the arithmetic side you subtract polynomials without losing a sign, multiply a binomial by a trinomial and combine the six products, and match coefficients in an identity that holds for every value of x to find unknown constants. On the factoring side you recognize a difference of squares with coefficients and two variables, factor completely by taking out the greatest common factor first, factor a trinomial whose leading coefficient is not 1 and check the result by expanding, and identify the greatest common factor of terms with two variables.

The factored form then does real work. You read the zeros of a function such as (x − 3)(2x + 5)(x + 1), decide which formula fits a cubic graph drawn on a coordinate grid from its x-intercepts, and use the fact that x − c is a factor exactly when the polynomial is zero at c. Two applications show why the structure matters: the extra area of a rectangular garden compared with a square patio, and a quick mental computation of 51² − 49² with the difference of squares.

The quiz has twelve multiple-choice questions whose wrong options are the typical slips — a sign not distributed, a factorization that is equivalent but not complete, a zero with the wrong sign. The flashcards list the identities and checks to know by heart. The printable written work asks you to simplify, factor and explain each step in your own words, and the oral exam lets you factor one expression at a time while an examiner asks how you checked it.

The content is based on the College Board's published skill descriptions for the PSAT/NMSQT Math section (Advanced Math: equivalent expressions; nonlinear functions). It is independent practice, not produced or endorsed by the College Board.

  • Add, subtract and multiply polynomials accurately
  • Factor completely using a common factor, a difference of squares and trinomials with any leading coefficient
  • Find unknown constants in a polynomial identity by matching coefficients
  • Read the zeros and x-intercepts of a polynomial function from its factored form
  • Use the link between a factor x − c and a zero at c

Practice material written by Zestly, based on the College Board's skill descriptions for the digital PSAT/NMSQT Math section (Advanced Math: equivalent expressions — factoring limited to a common factor, a difference of two squares and trinomials as a product of two binomials; adding, subtracting and multiplying polynomials; nonlinear functions — connections between the algebraic form and the graph of a polynomial function), as published in the Assessment Framework for the Digital SAT Suite.

Sample question

The graph of a cubic function y = f(x) on a coordinate grid: it comes up from the bottom left, crosses the x-axis at the marked points (−2, 0), (0, 0) and (4, 0), and rises to the top right.

The graph shows $y = f(x)$, where $f$ is a cubic function whose graph crosses the $x$-axis at the marked points $(-2, 0)$, $(0, 0)$ and $(4, 0)$. Which of the following could define $f$?

See the answer

$f(x) = x(x - 4)(x + 2)$

A zero at $x = r$ comes from a factor $x - r$: the zeros $-2$, $0$ and $4$ give the factors $x + 2$, $x$ and $x - 4$, so $f(x) = x(x - 4)(x + 2)$ fits (its leading coefficient is positive, so the graph rises to the right). $x(x + 4)(x - 2)$ has zeros $-4$, $0$ and $2$; $(x - 4)(x + 2)$ has no zero at $0$; and $x^2(x - 4)(x + 2)$ is not cubic and only touches the $x$-axis at $0$ without crossing it.

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