The Rectangle of Algebra
Expand (x+3)(x+5) by splitting it into four parts. No magic, just multiplication.
Equation
y = (x + 3)*(x + 5) - (x^2 + 0*x + 3*5)
Graph
Table
| x | y |
|---|---|
| -5 | -40 |
| -4 | -32 |
| -3 | -24 |
| -2 | -16 |
| -1 | -8 |
| 0 | 0 |
| 1 | 8 |
| 2 | 16 |
| 3 | 24 |
| 4 | 32 |
| 5 | 40 |
What this lesson covers
What you do
You shape the function y = (x + a)*(x + b) - (x^2 + k*x + a*b) and watch the graph answer.
Challenges to clear
- Expand (x+3)(x+5): keep a = 3, b = 5 and slide k to the middle coefficient (3 + 5). The difference must be 0 at x = 2.
- True for ALL x: still 0 at x = −1 — that makes it an identity.
- Your own brackets this time: (x + 4)(x + 6). Set a = 4 and b = 6.
- Now the middle coefficient — the SUM 4 + 6, while the constant is the PRODUCT 24. Slide k until the difference is 0 at x = 2.
Check yourself
In (x+3)(x+5) = x² + 8x + 15, where does the 8 come from?
And where does the 15 come from?
- 3 + 5 — the outer and inner products add up. — correct
- 3 × 5 — the product of the two numbers.
- 8 is just x's coefficient by convention.
- From doubling the 4 in x + 4.
- 3 × 5 — the product of the last terms. — correct
- 3 + 5 + 7.
- x times 15.
- It is a remainder.
Think about it
- In (x+3)(x+5) = x² + ?·x + 15 — what number multiplies x?