수포자 속출하는 ‘곱셈공식’ 절대 안 까 까먹는 확실한 방법 I 수학 기초 끝내기 I #정승제의50일수학

Main ideas / lessons

• Students commonly memorize “multiplication formulas” incorrectly—treating the signs (+/−) as if they just carry over unchanged—then get stuck later when expressions become more complex (square roots, factoring, complex numbers).
• The key fix is to internalize what the formula is doing structurally: it comes from distribution (expanding) and “collecting like terms,” not from memorizing sign-stays-the-same rules.
• The core focus is the perfect square / sum-and-difference patterns for squaring binomials of the form ((a \pm b)^2), and how to apply them with correct signs.
• The speaker also distinguishes between:
  • Cases where a shortcut formula exists (e.g., ((a\pm b)^2))
  • Cases where no such formula shortcut should be used, and expansion/multiplication must be done directly (especially when terms don’t match the “clean” binomial-square pattern, or when coefficients are not 1 in ways that break the simple perfect-square templates).

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Methodology / step-by-step instructions (as taught)

1) Correct way to think about the perfect square formula (derivation mindset)

• Don’t memorize “the + sign stays +” and “the − sign stays −.”
• Instead, remember it behaves like distribution:
  • Multiplying two binomials expands into multiple products (four total for ((a+b)^2)).
  • Then combine like terms (e.g., (a+a = 2a)).
• The point: the formula is a compressed form of distributing and then simplifying.

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2) How to square a binomial of the form ((a+b)^2) or ((a-b)^2)

Treat it as a single unified formula, not two separate ones.

For ((a+b)^2)

1. Square the first term: (a^2)
2. Double the product of the two terms: (+\,2ab)
3. Square the second term: (+\,b^2)

Result: [ (a+b)^2 = a^2 + 2ab + b^2 ]

For ((a-b)^2)

1. Square the first term: (a^2)
2. Double the product, but with a minus: (-\,2ab)
3. Square the second term (sign becomes positive when squaring): (+\,b^2)

Result: [ (a-b)^2 = a^2 - 2ab + b^2 ]

• Speaker’s “memory feeling”/mnemonic:
  • “Square the first.”
  • “Then square the first and second interaction with doubling.”
  • “Then square the second.”
  • Signs are determined by the pattern of the middle term ((\pm 2ab)), not by naive “sign carryover.”

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3) Example guidance: squaring a binomial with negatives / coefficients

• When the expression includes a negative in a term (e.g., (x-2y)):

  • You still follow the ((a-b)^2) logic:
    • First term squared
    • Middle term uses doubling and the sign pattern
    • Second term squared becomes positive because ((-2y)^2 = 4y^2)

• Another example mentioned: squaring ((-3 + 2x))

  • Square the first term: ((-3)^2 = 9)
  • Compute the doubled product part with the correct sign
  • Square the second term: ((2x)^2 = 4x^2)
  • Combine to get the accepted final expression.

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4) Emphasis: “Sum and difference” is the concept name

• The speaker stresses the terminology: it’s called sum and difference (linked to ((a+b)^2) and ((a-b)^2)), not “subtraction twice” as a naive sign rule.
• The “transfer” idea:
  • Even when the two binomials look similar except for visible signs, the formula changes only in the middle term according to the (\pm) structure.

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5) When you should NOT use a “shortcut formula”

The speaker warns that some expressions are not cleanly handled by the perfect-square templates.

• If the task involves expressions that don’t fit the “simple” binomial-square pattern (especially when trying to treat something as a formula when it isn’t one):
  • No perfect shortcut exists
  • You must expand directly (multiply term-by-term “one by one”)

Instruction style for the non-formula cases:

• Expand/multiply sequentially
• Combine like terms along the way (the speaker suggests not waiting until the very end)
• Multiply constants at the end

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6) Example method for direct expansion (illustrative)

• For linear-term multiplications after expansion:
  • Multiply each part in order
  • Combine like terms immediately (e.g., when (-3x) interacts with something producing (-6x) and (+x), combine to get (-5x))
• For cases including several linear parts and constants:
  • Add/subtract the coefficients after multiplying
  • Multiply constants last

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Speakers / sources featured

• 정승제 (Jeong Seung-je / “정승제”) — the teacher/host (also referenced via the channel/title: “#정승제의50일수학”)