The standard formula for the cube of a sum is:
This can also be expressed in its fully expanded form by distributing the $3ab$:
The image provides a clear algebraic proof for why this identity works by breaking down the exponent:
When you expand $(a+b)^3$, notice the pattern of the exponents:
The powers of $a$ decrease: $a^3, a^2, a^1$.
The powers of $b$ increase: $b^1, b^2, b^3$.
The middle coefficients are both 3.
The identity for the cube of a sum can be expressed in two ways:
Factored form: $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$
Expanded form: $a^3 + b^3 + 3a^2b + 3ab^2$
To prove this, we break the cube into a linear term and a squared term:
By substituting the identity for $(a + b)^2$ and distributing the terms, we get:
Grouping the like terms gives us the final identity.
Similar to the sum, the difference identity is:
Factored form: $(a – b)^3 = a^3 – b^3 – 3ab(a – b)$
Expanded form: $a^3 – b^3 – 3a^2b + 3ab^3$
Note on Proof: This can be derived by replacing $b$ with $(-b)$ in the first identity. Since $(-b)^3$ is negative and $(-b)^2$ is positive, the signs in the expansion alternate.
Using the sum identity:
Cube the terms: $(2x)^3 + (3y)^3 = 8x^3 + 27y^3$
Apply the middle term: $3(2x)(3y)(2x + 3y) = 18xy(2x + 3y)$
Final Expansion: $8x^3 + 27y^3 + 36x^2y + 54xy^2$
Using the difference identity:
Cube the terms: $(3x)^3 – (2y)^3 = 27x^3 – 8y^3$
Apply the middle term: $-3(3x)(2y)(3x – 2y) = -18xy(3x – 2y)$
Final Expansion: $27x^3 – 8y^3 – 54x^2y + 36xy^2$
| Expression | Expanded Identity |
| $(a + b)^3$ | $a^3 + 3a^2b + 3ab^2 + b^3$ |
| $(a – b)^3$ | $a^3 – 3a^2b + 3ab^2 – b^3$ |
The identity for the cube of a sum can be expressed in two primary forms:
Factored Form: $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$
Expanded Form: $a^3 + b^3 + 3a^2b + 3ab^2$
To prove this, we treat the cube as a product of a linear term and a squared term:
By substituting the identity for $(a+b)^2$, we get $(a + b)(a^2 + b^2 + 2ab)$. Multiplying these terms out results in:
When working with a subtraction sign, the identity shifts slightly:
Factored Form: $(a – b)^3 = a^3 – b^3 – 3ab(a – b)$
Expanded Form: $a^3 – b^3 – 3a^2b + 3ab^2$
Example: To expand $(3x – 2y)^3$, we apply the identity:
Sometimes you need to work backward from the sum of two cubes. This identity is derived from the first:
Identity: $a^3 + b^3 = (a + b)^3 – 3ab(a + b)$
Alternative Form: $(a + b)(a^2 – ab + b^2)$
Problem: If $a + b = 10$ and $ab = 21$, find the value of $a^3 + b^3$.
Solution:
Substitute values into the identity: $(10)^3 – 3(21)(10)$
Simplify: $10[ (10)^2 – 3(21) ]$
Calculate: $10(100 – 63) = 10(37) = 370$
| Expression | Identity Formula |
| $(a + b)^3$ | $a^3 + b^3 + 3a^2b + 3ab^2$ |
| $(a – b)^3$ | $a^3 – b^3 – 3a^2b + 3ab^2$ |
| $a^3 + b^3$ | $(a + b)(a^2 – ab + b^2)$ |
The identity for the cube of a sum can be expressed in two primary forms:
Factored Form: $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$
Expanded Form: $a^3 + b^3 + 3a^2b + 3ab^2$
To prove this, we treat the cube as a product of a linear term and a squared term:
By substituting the identity for $(a+b)^2$, we get $(a + b)(a^2 + b^2 + 2ab)$. Multiplying these terms out results in:
When working with a subtraction sign, the identity shifts slightly:
Factored Form: $(a – b)^3 = a^3 – b^3 – 3ab(a – b)$
Expanded Form: $a^3 – b^3 – 3a^2b + 3ab^2$
Example: To expand $(3x – 2y)^3$, we apply the identity:
These identities allow you to factor or find the value of expressions involving two separate cubic terms.
Identity: $a^3 + b^3 = (a + b)^3 – 3ab(a + b)$
Alternative Form: $(a + b)(a^2 – ab + b^2)$
Practical Application: If $a + b = 10$ and $ab = 21$, find the value of $a^3 + b^3$:
Substitute values into the identity: $(10)^3 – 3(21)(10)$
Factor out the 10: $10[ (10)^2 – 3(21) ]$
Calculate: $10(100 – 63) = 10(37) = 370$
Identity: $a^3 – b^3 = (a – b)^3 + 3ab(a – b)$
Alternative Form: $(a – b)(a^2 + ab + b^2)$
| Expression | Identity Formula |
| $(a + b)^3$ | $a^3 + b^3 + 3a^2b + 3ab^2$ |
| $(a – b)^3$ | $a^3 – b^3 – 3a^2b + 3ab^2$ |
| $a^3 + b^3$ | $(a + b)(a^2 – ab + b^2)$ |
| $a^3 – b^3$ | $(a – b)(a^2 + ab + b^2)$ |