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MathChapter 10: Law of Exponents and Polynomials

Notes

FOIL Method and Special Products Notes

Multiply binomials through complete distribution and recognize three high-value special-product identities.

Practice this topic
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About 54 minutes
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FOIL is a memory aid for distributing every term of one binomial across every term of another. Special products are not separate tricks: they are frequently recurring FOIL results whose structure lets you work faster and check signs.

Learning objectives

  • Connect FOIL to the Distributive Property.
  • Expand any two binomials and combine their middle terms.
  • Recognize squares of sums, squares of differences, and conjugate products.
  • Use reverse recognition to identify an unexpanded product.
  • Avoid omitted middle terms and sign errors.

FOIL as organized distribution

FOIL is organized distribution

Diagram showing first, outer, inner, and last products when multiplying two binomials.

\[(3x+4)(2x-5)\]

  1. First\((3x)(2x)\)\(6x^2\)
  2. Outer\((3x)(-5)\)\(-15x\)
  3. Inner\((4)(2x)\)\(8x\)
  4. Last\((4)(-5)\)\(-20\)

Expanded: \(6x^2-15x+8x-20\)

Combined: \(6x^2-7x-20\)

Multiply two binomials

  1. First

    Multiply the first terms.

  2. Outer and inner

    Compute both cross-products and preserve their signs.

  3. Last

    Multiply the constants or last terms.

  4. Combine

    Add the two like middle terms.

Special-product patterns and sign structure
PatternExpansionRecognition clueCommon trap
Square of a sum\((a+b)^2=a^2+2ab+b^2\)Positive middle termOmitting \(2ab\)
Square of a difference\((a-b)^2=a^2-2ab+b^2\)Negative middle; positive lastWriting \(-b^2\)
Sum times difference\((a+b)(a-b)=a^2-b^2\)Conjugates; middle terms cancelWriting \(a^2+b^2\)

Work with structure before arithmetic

Worked example

Square a binomial

Expand \((3x-5)^2\).

  1. Identify \(a\) and \(b\)

    Use \(a=3x\) and \(b=5\) in \((a-b)^2\).

  2. Apply all three terms

    Compute \((3x)^2-2(3x)(5)+5^2\).

The expansion is \(9x^2-30x+25\).
Mini check

Recognize a conjugate product

Expand \((4y+7)(4y-7)\).

  1. \(16y^2-49\)
  2. \(16y^2+49\)
  3. \(16y^2-56y+49\)
Show answer and explanation

Answer: \(16y^2-49\)

The factors are conjugates, so the cross-products cancel and \(a^2-b^2\) remains.

Key takeaways

Key takeaways

What to remember

  • FOIL is complete distribution.
  • A squared binomial contains a middle term \(\pm2ab\).
  • Conjugates produce a difference of squares.
  • Check patterns by expanding when uncertain.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.