Cube of a Binomial
Cube of a binomial is a Grade 7 algebra skill from Yoshiwara Intermediate Algebra where students expand expressions of the form (a + b)^3 or (a - b)^3 using the binomial cube formula. The result is a^3 + 3a^2b + 3ab^2 + b^3, which follows a predictable pattern.
Key Concepts
Property 1. $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$ 2. $(x y)^3 = x^3 3x^2y + 3xy^2 y^3$.
Examples To expand $(a + 4)^3$, use the first formula with $x=a$ and $y=4$: $a^3 + 3(a)^2(4) + 3(a)(4)^2 + 4^3 = a^3 + 12a^2 + 48a + 64$. To expand $(2b 1)^3$, use the second formula with $x=2b$ and $y=1$: $(2b)^3 3(2b)^2(1) + 3(2b)(1)^2 1^3 = 8b^3 12b^2 + 6b 1$. To expand $(z^2 + 3)^3$, use the first formula with $x=z^2$ and $y=3$: $(z^2)^3 + 3(z^2)^2(3) + 3(z^2)(3)^2 + 3^3 = z^6 + 9z^4 + 27z^2 + 27$.
Explanation These formulas are shortcuts for expanding expressions like $(a+b)^3$ without performing a lengthy multiplication. Memorizing these patterns for the sum and difference of two cubed terms saves time and helps prevent common algebra mistakes.
Common Questions
What is the formula for the cube of a binomial?
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 and (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
How do you expand (x + 2)^3?
Using (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 with a = x, b = 2: x^3 + 3x^2(2) + 3x(4) + 8 = x^3 + 6x^2 + 12x + 8.
Why should you use the formula instead of multiplying?
The formula is faster and less error-prone than multiplying (a + b)(a + b)(a + b) step by step.
How does the cube of a binomial relate to Pascal triangle?
The coefficients 1, 3, 3, 1 in the cube of a binomial correspond to the third row of Pascal triangle.