Multiply Monomials: The 3-Variable Swap
See how coefficients multiply and variables group when expanding a product.
Equation
y = (1*x^2) * (1*x)
Graph
Table
| x | y |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
What this lesson covers
What you do
You shape the function y = (a*x^2) * (b*x) and watch the graph answer.
Challenges to clear
- Build (2x^2)(3x): set a = 2 and b = 3. At x = 2 you should see 48 — that is 6x^3 = 6 × 8.
- Check the coefficient alone: at x = 1 the product is just a × b = 6.
- Higher powers now: (a x³)(b x²). Set a = 2.
- Slide b until x = 2 reads 256. Powers ADD to give x⁵, coefficients MULTIPLY — 8 × 32 = 256.
Check yourself
Why must the variable base (x) be the same to add the exponents?
Evaluate (2x^2) * (3x^4) using the law.
- Because exponent laws only apply when bases are identical; otherwise, you cannot combine them into a single power. — correct
- Because adding exponents is the same as multiplying bases.
- Because different variables represent different quantities that cannot be mathematically combined.
- Because the coefficients a and b determine if the exponents can be added.
- 6x^6 — correct
- 5x^6
- 6x^8
- 5x^8
Think about it
- (2x^2)(3x) groups as (2×3) × (x^2 · x). What are the new coefficient and exponent?