Quick Answer
Factoring is the mathematical process of breaking down a polynomial into a product of simpler expressions or factors. On the Digital SAT, this technique is frequently tested in the Math modules, appearing in approximately 10-15% of algebra and advanced math questions, often requiring students to identify equivalent expressions or find the zeros of quadratic functions.
Factoring involves rewriting a mathematical expression, such as a quadratic ax² + bx + c, as the product of its constituent factors like (x - p)(x - q). It is the inverse operation of expansion, used primarily to simplify equations and solve for unknown variables.
Question: Which of the following is an equivalent form of the expression x² - 9x + 20? A) (x - 4)(x - 5) B) (x + 4)(x + 5) C) (x - 2)(x - 10) D) (x + 2)(x + 10) Solution: To factor x² - 9x + 20, find two numbers that multiply to 20 and add to -9. These numbers are -4 and -5. Thus, the factored form is (x - 4)(x - 5). Choice A is correct.
Sign errors: Students often confuse positive and negative signs when factoring trinomials, especially when the constant term is positive but the middle term is negative.
Forgetting the GCF: Many test-takers jump straight into trinomial factoring without first checking if a greatest common factor can be pulled out, making the numbers much harder to manage.
Incomplete factoring: Students sometimes stop after one step, such as factoring out a variable, without realizing the remaining expression (like a difference of squares) can be factored further.
Students targeting 750+ should know that the SAT often uses factoring in 'structure' questions where you must identify which form of an equation reveals the coordinates of the vertex or the x-intercepts without further calculation. Mastering the difference of squares and perfect square trinomial patterns can save crucial seconds compared to using the quadratic formula.
Expression
An expression is a mathematical phrase combining numbers, variables, and operators without an equals sign. On the Digital SAT, expressions are fundamental to the Algebra and Advanced Math sections. Typically, approximately 20-30% of Math questions involve manipulating or simplifying expressions, appearing frequently in both multiple-choice and student-produced response formats.
FOIL Method
The FOIL Method is a mnemonic used on the Digital SAT to multiply two binomials effectively. This algebraic technique is essential for expanding expressions in Math Modules 1 and 2. Typically, students encounter 1 to 3 questions per test requiring binomial expansion or its reverse, factoring, to solve quadratic equations and identify equivalent expressions.
Polynomial
A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents. On the Digital SAT, polynomials frequently appear in the Advanced Math section, typically requiring students to add, subtract, multiply, or factor expressions. These questions often represent approximately 10-15% of the math content across both modules.
Quadratic Equation
A quadratic equation is a second-degree polynomial equation typically written in standard form as ax² + bx + c = 0. On the Digital SAT, these equations appear frequently in the Advanced Math section, accounting for approximately 15% of math questions. Students must solve them using factoring, completing the square, or the quadratic formula.
Zeros of a Function
Zeros of a function are the input values where f(x) equals zero. On the Digital SAT, this concept is a staple of the Advanced Math section, typically appearing in 3–5 questions per exam. Students must identify these as x-intercepts on a graph or solve for them algebraically using factoring or the quadratic formula.
Factoring on the SAT is the process of decomposing algebraic expressions into products of simpler terms to solve equations or identify equivalent forms. It is a fundamental skill within the Heart of Algebra and Passport to Advanced Math domains. Students are typically expected to factor quadratics to find x-intercepts or to simplify complex rational expressions during both modules of the Math section.
To calculate factors of a quadratic expression like ax² + bx + c, you look for two numbers that multiply to the product of 'a' and 'c' and add to 'b'. Once these numbers are found, you rewrite the middle term and use grouping, or use the shortcut for a=1 by placing the numbers directly into (x + p)(x + q). For binomials, always check for a greatest common factor first.
Factoring and the FOIL method are inverse operations of one another. While the FOIL method (First, Outer, Inner, Last) is used to multiply two binomials to create a simplified polynomial, factoring takes that polynomial and breaks it back down into its original binomial components. On the SAT, you use FOIL to expand expressions and factoring to solve for roots or find intercepts.
Approximately 10% to 15% of the Math questions on a typical Digital SAT will involve factoring in some capacity. While some questions explicitly ask for the factored form of an expression, others require factoring as a preliminary step to solve for variables, find the zeros of a function, or simplify rational expressions. It is one of the most consistently tested algebraic skills on the exam.
