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➕ Algebra 1 · Exponents and Polynomials

Special Products of Binomials: notes and common mistakes

Study notes

Key ideas

  • (a + b)2 = a2 + 2ab + b2 and (a − b)2 = a2 − 2ab + b2. Square the first term, double the product, square the last term. When the first term has no minus sign in front of it, the middle term takes the sign between the two terms. The last term b2 is never negative.
  • The patterns hold for negative a and b too. A first term with a minus sign in front of it can make the middle term negative even when the sign between the terms is a plus: (−2x + 5)2 = 4x2 − 20x + 25. Treat the binomial as a sum of two signed terms and double the product of those terms.
  • (a + b)(a − b) = a2 − b2: the middle terms cancel. It needs the same two terms in both factors, with opposite signs between them.
  • (a + b)2 ≠ a2 + b2 unless a = 0 or b = 0. The missing piece is the cross term 2ab.
  • a and b can be any terms, like 3x, x2 or 2y. Square the whole term: (3x)2 = 9x2.
  • These are identities, true for every value. Multiplying out with letters proves them. Testing a few numbers does not.

Steps

  1. Decide which pattern fits: a binomial squared, a sum times a difference of the same two terms, or neither.
  2. Name a and b. Take a as the first term exactly as written, including its sign. Take b as the second term without the sign between the two terms. If the sign between them is a plus, use a2 + 2ab + b2. If it is a minus, use a2 − 2ab + b2. For a sum times a difference, a is the term whose sign stays the same and b is the term whose sign flips, and the answer is a2 − b2.
  3. Do not count a minus sign twice. Either use the minus pattern with a positive b, or write a − b as a + (−b) and use the plus pattern with a negative b. Examples: (2x − 7)2 has a = 2x and b = 7, and (−2x + 5)2 has a = −2x and b = 5.
  4. Write the pattern with a and b in parentheses, then simplify each part completely: (3x)2 = 9x2.
  5. Check with a test value such as x = 3.

Formulas

  • Square of a sum: (a + b)2 = a2 + 2ab + b2
  • Square of a difference: (a − b)2 = a2 − 2ab + b2
  • Sum times difference: (a + b)(a − b) = a2 − b2
  • Squares that differ: (a + b)2 − (a − b)2 = 4ab
  • Mental-math product: (m − d)(m + d) = m2 − d2

Watch out

  • (a + b)2 ≠ a2 + b2. Write the repeated factor first: (a + b)(a + b).
  • (a − b)2 ≠ a2 − b2. The expression a2 − b2 is the answer to (a + b)(a − b).
  • The last term of (a − b)2 is +b2, not −b2, because (−b)(−b) = b2.
  • Square the whole term: (3x)2 = 9x2, not 3x2.
  • A shortcut needs its exact shape. (x + 3)(x − 5) is neither a square nor a sum times a difference of the same two terms.
  • A few matching numbers do not prove an identity.

Why it works

  • Why 2ab? Multiplying (a + b)(a + b) gives two middle products, ab and ba (the Outer and Inner products). They are equal, so they combine into 2ab. In the picture they are the two rectangles.
  • Why no middle term in (a + b)(a − b)? The Outer and Inner products are −ab and +ab, which cancel.
  • Why learn the patterns? They save time and make mental math easy. They also help you see structure: x4 − 16 is (x2)2 − 42, a difference of squares. Read backward, the patterns become the factoring shortcuts you will use in the factoring unit.

Common mistakes

✗ Mistake(a + b)2 = a2 + b2 (squaring each piece).
✓ InsteadSquaring multiplies the whole sum by itself: (a + b)(a + b) = a2 + 2ab + b2. The cross term 2ab is part of the answer. For a = 2 and b = 3, (2 + 3)2 = 25 but 22 + 32 = 13.
✗ Mistake(x − 13)2 = x2 − 169 (the square of a difference is a difference of squares).
✓ Insteadx2 − 169 is the answer to (x + 13)(x − 13). Squaring the difference gives (x − 13)2 = x2 − 26x + 169.
✗ Mistake(x − 13)2 = x2 − 26x − 169 (the last term keeps the minus sign).
✓ InsteadThe last term is (−13)2 = +169. Only the middle term is negative: x2 − 26x + 169.
✗ MistakeIn (−2x + 5)2 the sign between the terms is a plus, so every term of the answer is positive.
✓ InsteadThe first term is −2x, so the double product is 2 × (−2x) × 5 = −20x. The answer is 4x2 − 20x + 25.
✗ Mistake(a + b)(a − b) is the same thing as (a − b)2, so it has a middle term.
✓ InsteadThey are different. (a − b)2 = a2 − 2ab + b2 has a middle term, but in (a + b)(a − b) the middle terms −ab and +ab cancel, giving a2 − b2.
✗ Mistake(3x + 2)2 = 3x2 + 12x + 4 (not squaring the coefficient).
✓ InsteadSquare the whole first term: (3x)2 = 9x2. So (3x + 2)2 = 9x2 + 12x + 4.
✗ Mistake(x + 2)(x − 9) is a difference of squares, so it equals x2 − 18.
✓ InsteadThe pattern needs the same two terms in both factors. Here 2 and 9 differ, so the middle terms do not cancel: (x + 2)(x − 9) = x2 − 7x − 18.

Teach it back

Explain to a friend (or a pet!) why (n + 6)2 is not n2 + 36, how to expand it, and what the picture of the square shows.

Explaining it out loud shows what you really know. Did you…

  • I wrote (n + 6)2 = (n + 6)(n + 6) and multiplied every term by every term.
  • I got n2 + 12n + 36 and said where the 12n comes from: 2 × n × 6, the two rectangles in the picture.
  • I showed with n = 3 that (3 + 6)2 = 81 but 32 + 36 = 45, so n2 + 36 cannot be right.

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  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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