[Common Math 1 Part 5] Mastering Multiplication Identities

Instead of memorizing formulas, understand them as stamps created by multiplication and overlaid to accumulate coefficients.

1. Same Operation, Different Packaging

As we saw earlier, every multiplication eventually boils down to

multiply every pair of terms, then
add the ones that share the same pair of exponents.

Multiplication identities are no exception.

2. Grade 10 Must-Know Identities

2‑1. Perfect-square identities

(a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

Watch the middle term:

$(a + b)^2$ creates $ab$ twice → $+2ab$ .
$(a - b)^2$ also creates $ab$ twice but each has a minus sign → $-2ab$ .

2‑2. Sum-and-difference

(a + b)(a - b) = a^2 - b^2

The middle $+ab$ and $-ab$ cancel each other out.

2‑3. Quadratic-focused identities

(x + a)(x + b) = x^2 + (a + b)x + ab

(ax + b)(cx + d) = acx^2 + (ad + bc)x + bd

From a coefficient viewpoint:

The coefficient of $x$ becomes $a + b$ .
The constant term becomes $ab$ .

3. The Stamp Model Behind Every Identity

All identities share a single rule. Throughout this article the coordinate system is row = $y$ , column = $x$ .

Use each term of one expression to scale the entire other expression and create a stamp.
Shift the stamp by the appropriate degree and overlay it.
Add overlapping cells.

This explains why

certain terms overlap twice, giving coefficient 2, and
others cancel because they carry opposite signs.

That overlap/cancel behavior is precisely why the identities hold.

4. Check the Identities With Stamp Visualizations

Pick a preset below:

Example 1: $x + y$ and $x + y$ → $(x + y)^2$
Example 2: $x - y$ and $x - y$ → $(x - y)^2$
Example 3: Choose $(x + y)^3$ to watch the square followed by an extra multiplication.

2변수 다항식 곱(도장 겹치기)

도장 생성 -> 겹치기 -> 누적 덧셈

적용 0 / 2

P(x,y)=x+y

Q(x,y)=x-y

현재 공식: 직접 입력

곱셈공식 프리셋

중3 필수

공통수학1

도장 목록

Q의 0이 아닌 성분으로부터 생성된 도장 2개

원래 Q 행렬과 선택 위치

y^{1}x^{1}

y^{1}x^{0}

-1

y^{0}x^{1}

y^{0}x^{0}

현재 선택된 Q 성분이 P의 이동 위치를 결정합니다.

x/y

x^{2}

x^{1}

x^{0}

y^{2}

y^{1}

y^{0}

현재 연산과 최종 결과

모든 도장 적용 완료 (최종 결과 행렬)

5. Practical Checkpoints

Before memorizing, look at why the middle terms appear or disappear.
Most sign mistakes happen in those middle terms.
Always combine like terms at the end of expansion.

6. Key Takeaways

Multiplication identities are just repeated applications of the standard expansion rule.
From the stamp (convolution) viewpoint you immediately see where coefficients of 2 or 0 arise.
When memorizing, keep the “middle-term creation/cancellation” story in mind to avoid errors.