Cube of Sum Identity with Expansion and Examples
Cube of Sum Identity gives the expansion of the cube of two added quantities. The formula (a + b)³ = a³ + 3a²b + 3ab² + b³ helps simplify algebraic expressions, calculate numerical cubes and solve quantitative aptitude problems without lengthy multiplication.
What Is the Cube of Sum Identity?
Here, a and b can be numbers, variables or algebraic expressions. The coefficients of the middle terms are both 3, and every term has a total degree of 3. For example, (x + 2)³ = x³ + 6x² + 12x + 8.
Cube of Sum Identity Video Lessons
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Cube Identities: A Plus or Minus B
Learn the cube identities for (A+B)³ and (A−B)³, along with the factorisations of A³+B³ and A³−B³ for solving algebraic expressions.
Cube of Sum Identity Formula & Tricks
Important Formulas
Expand the cube of a sum into the cube of the first term, two middle product terms and the cube of the second term.
Since (a + b)² = a² + 2ab + b², multiplying this square by (a + b) gives the cube identity.
The coefficients 1, 3, 3, 1 come from the third row of Pascal's triangle.
Quick Tricks
For a cube of a sum, write the powers of the first term in descending order and the powers of the second term in ascending order. Apply coefficients 1, 3, 3 and 1.
When direct expansion is inconvenient, calculate (a + b)² first and multiply it by (a + b). This reduces sign and power errors.
Cube of Sum Identity Concepts
Expanding a Cube of Two Terms
The first term is the cube of a, the second and third terms contain the coefficient 3, and the last term is the cube of b. The powers of a decrease from 3 to 0, while the powers of b increase from 0 to 3.
Applying the Identity to Numerical Values
Choose a and b so that a³, b³ and the product terms are easy to calculate. For 12³, take a = 10 and b = 2.
Expanding Expressions with Terms
Substitute the complete expressions for a and b, then simplify each term carefully. For example, in (x + 2y)³, take a = x and b = 2y.
Recognising a Cube of Sum
The first and last terms must be perfect cubes, and the middle terms must have coefficients 3a²b and 3ab². For x³ + 6x² + 12x + 8, the terms match a = x and b = 2, so the expression is (x + 2)³.
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Cube of Sum Identity Revision Points
Use these facts to recall the expansion and apply it correctly.
- (a + b)³ = a³ + 3a²b + 3ab² + b³.
- The coefficient pattern is 1, 3, 3, 1.
- The powers of the first term decrease from 3 to 0.
- The powers of the second term increase from 0 to 3.
- For (a + b)³, all terms have positive signs when a and b are positive algebraic terms.
- The identity applies to numbers, variables and complete algebraic expressions.
Cube of Sum Identity FAQs
What is the cube of sum identity formula?
The formula is (a + b)³ = a³ + 3a²b + 3ab² + b³.
How do you expand (x + 3)³?
(x + 3)³ = x³ + 3x²(3) + 3x(3²) + 3³ = x³ + 9x² + 27x + 27.
What is 11³ using the cube of sum identity?
11³ = (10 + 1)³ = 1000 + 300 + 30 + 1 = 1331.
How is the cube of sum different from the cube of difference?
The cube of sum is (a + b)³ = a³ + 3a²b + 3ab² + b³. The cube of difference is (a - b)³ = a³ - 3a²b + 3ab² - b³.
How can you identify whether an expression is a cube of a sum?
Check whether it has the form a³ + 3a²b + 3ab² + b³. For example, x³ + 6x² + 12x + 8 equals (x + 2)³.
What is the expansion of (2a + 3b)³?
(2a + 3b)³ = 8a³ + 36a²b + 54ab² + 27b³.