The Cube of a Sum
Expand (a + b)^3 step-by-step. Watch the identity emerge from multiplication.
Equation
y = (x + 2)^3 - (x^3 + 0*x^2*2 + 3*x*2^2 + 2^3)
Graph
Table
| x | y |
|---|---|
| -5 | 150 |
| -4 | 96 |
| -3 | 54 |
| -2 | 24 |
| -1 | 6 |
| 0 | 0 |
| 1 | 6 |
| 2 | 24 |
| 3 | 54 |
| 4 | 96 |
| 5 | 150 |
What this lesson covers
What you do
You shape the function y = (x + a)^3 - (x^3 + k*x^2*a + 3*x*a^2 + a^3) and watch the graph answer.
Challenges to clear
- (x+a)³ needs a 3x²a term: slide k until the difference is 0 at x = 2.
- Identity: 0 at x = −1 as well — the expansion is x³ + 3x²a + 3xa² + a³.
- Now the cube of a DIFFERENCE: (x − a)³. Set a = 4.
- The signs alternate, but the coefficients are still 1, 3, 3, 1. Slide k until the difference is 0 at x = 2.
Check yourself
Why does the graph staying flat at y=0 prove the identity?
What is the coefficient of the x*a^2 term in the expansion of (x+a)^3?
If a=2, what is the constant term (a^3) in the expansion?
- Because the graph is a straight line.
- Because y = LHS - RHS, so y=0 means LHS = RHS for all x and a. — correct
- Because the coefficients are all positive.
- Because we only checked x=2.
- 1
- 2
- 3 — correct
- 4
- 4
- 6
- 8 — correct
- 12