Factor polynomials (GCF, difference of squares, trinomials) and expand products. | Equivalent expressions | SAT Math

Factoring rewrites an expression as a product. The SAT primarily tests three types: pulling out a GCF, recognizing a difference of squares, and factoring trinomials.

核心知识

Always check for a GCF first. Then look at the number of terms: two terms suggests difference of squares $𝑎 2 - 𝑏 2 = (𝑎 - 𝑏) (𝑎 + 𝑏)$ ; three terms suggests trinomial factoring where you find two numbers that multiply to $𝑐$ and add to $𝑏$ in $𝑥 2 + 𝑏 𝑥 + 𝑐$ .

深入理解

Start every factoring problem by asking: do all terms share a common factor? Pull it out first. This simplifies everything that follows and is easy to overlook under time pressure.

For two-term expressions, check whether both terms are perfect squares separated by a minus sign. If so, apply $𝑎 2 - 𝑏 2 = (𝑎 - 𝑏) (𝑎 + 𝑏)$ . A sum of squares $𝑎 2 + 𝑏 2$ does not factor over the reals—this is a common SAT trap.

For three-term expressions $𝑥 2 + 𝑏 𝑥 + 𝑐$ , find two numbers whose product is $𝑐$ and whose sum is $𝑏$ . When the leading coefficient isn't 1, factor out the GCF first if possible, or use the AC method.

Always verify your factored form by expanding it back out. This takes seconds and catches sign errors.

分步讲解

Pull out any GCF from all terms.
Count remaining terms: two → check difference of squares; three → try trinomial factoring.
For difference of squares: identify $𝑎$ and $𝑏$ where the expression equals $𝑎 2 - 𝑏 2$ , then write $(𝑎 - 𝑏) (𝑎 + 𝑏)$ .
For trinomials: find two numbers that multiply to the constant term and add to the coefficient of the middle term.
Write the factored form and verify by expanding.

常见误解

Attempting to factor $𝑎 2 + 𝑏 2$ as $(𝑎 + 𝑏) 2$ —a sum of squares doesn't factor over the reals.
Forgetting to take the square root of the coefficient in difference of squares: writing $(4 𝑥 - 5) (𝑥 + 5)$ instead of $(2 𝑥 - 5) (2 𝑥 + 5)$ for $4 𝑥 2 - 25$ .
Stopping after pulling out the GCF without checking whether the remaining expression factors further.

题目

示例解析

Which of the following is equivalent to $4 𝑥 2 - 25$ ?

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