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1.6.2 Expanding Expressions

Apply distributive property and FOIL method to expand polynomials and products of algebraic expressions.

定义

Expanding expressions refers to the process of applying the distributive property and algebraic multiplication techniques to rewrite products of algebraic expressions as a sum or difference of terms. The distributive property states that for any real numbers or algebraic expressions \(a\), \(b\), and \(c\): \(a(b + c) = ab + ac\). When expanding products of binomials or polynomials, we multiply each term in the first expression by each term in the second expression, then combine like terms. The FOIL method is a specific technique used for expanding the product of two binomials \((a + b)(c + d)\), where we multiply the First terms, Outer terms, Inner terms, and Last terms, then combine the results.

核心公式

  • \(a(b + c) = ab + ac\)
  • \((a + b)(c + d) = ac + ad + bc + bd\)
  • \((a + b)^2 = a^2 + 2ab + b^2\)
  • \((a - b)^2 = a^2 - 2ab + b^2\)
  • \((a + b)(a - b) = a^2 - b^2\)

易错点

  • ⚠️ Forgetting to distribute to all terms: Students often only multiply the first term of one expression by all terms of the other, neglecting to multiply the remaining terms. For example, writing \(2(x + 3) = 2x + 3\) instead of \(2x + 6\).
  • ⚠️ Sign errors when expanding: Failing to correctly handle negative signs, such as incorrectly expanding \((x - 3)(x + 2)\) as \(x^2 + 2x - 3x - 6 = x^2 - x - 6\) when the correct answer is \(x^2 - x - 6\) (though this example is correct, common errors include \((x - 3)^2 = x^2 - 9\) instead of \(x^2 - 6x + 9\)).
  • ⚠️ Not combining like terms after expansion: After expanding an expression, students may forget to simplify by combining like terms. For instance, expanding \((x + 2)(x + 3)\) to get \(x^2 + 3x + 2x + 6\) but leaving it unsimplified instead of combining to \(x^2 + 5x + 6\).
  • ⚠️ Misapplying special product formulas: Incorrectly using formulas like \((a + b)^2 = a^2 + b^2\) instead of the correct \(a^2 + 2ab + b^2\), or confusing the difference of squares formula with other patterns.