知识点 4

Factor and simplify polynomial expressions; apply identities (e.g., difference of squares).

Look for factoring patterns before you expand or simplify.

核心知识

Before expanding or doing long algebra, look for structure: a common factor, a trinomial pattern, or an identity such as 𝑎2 −𝑏2 =(𝑎 −𝑏)(𝑎 +𝑏).

深入理解

Rule: ACT polynomial questions reward recognition. If you notice a greatest common factor, a perfect-square trinomial, or a difference of squares early, the expression usually becomes short fast.

The most common identity to recognize is difference of squares. Whenever you see two squares being subtracted, factor instead of expanding: 𝑢2 −𝑣2 =(𝑢 −𝑣)(𝑢 +𝑣). That move shows up in factoring, solving quadratics, and simplifying later expressions.

分步讲解

  1. Check first for a greatest common factor.
  2. Then ask whether the expression matches a standard pattern such as difference of squares or a factorable trinomial.
  3. Factor completely before deciding whether anything else can simplify.

常见误解

  • Expanding first even when the expression is ready to factor.
  • Treating a difference of squares like a square trinomial.
  • Missing a greatest common factor before checking other patterns.
题目

示例解析

Which expression is equivalent to 49𝑚2 −81?

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