Sum of Cubes Formula
The formula to find the addition of two polynomials, a3 + b3 is known as the sum of cubes formula. Let's learn more about the sum of cubes formula with a few solved examples. This factoring formula comes in very handy when solving algebraic expressions of various types. Memorizing this formula is also easy and can be done within a matter of minutes. It is very similar to the difference in cubes formula as well.
What Is the Sum of Cubes Formula?
In this section, let us go further and understand what exactly does it mean when some is referring to the sum of cubes. The formula to the sum of cubes formula is given as:
a3 + b3 = (a + b)(a2 - ab + b2)
where,
- a is the first variable
- b is the second variable
Proof of Sum of Cubes Formula
To prove or verify that sum of cubes formula that is, a3 + b3 = (a + b) (a2 - ab + b2) we need to prove here LHS = RHS.
LHS term = a3 + b3
On Solving RHS term we get,
= (a + b) (a2 - ab + b2)
On multiplying the a and b separately with (a2 + ab + b2) we get
= a (a2 - ab + b2) + b(a2 - ab + b2)
= a3 - a2b + ab2 + a2b - ab2 + b3
= a3 - a2b + a2b + ab2- ab2 + b3
= a3 - 0 + 0 + b3
= a3 + b3
Hence proved, LHS = RHS
Examples on Sum of Cubes Formula
Example1: Use the sum of cubes formula to find the factor of 216x3 + 64.
To find: Factor of 216x3 + 64, using the sum of cubes formula.
216x3+ 64 = (6x)3 + 43
Using the sum of cubes formula,
a3 + b3 = (a + b)(a2 - ab + b2)
Put the values,
(6x)3 + 43 = (6x + 4)((6x)2 - 6x × 4 + 42)
(6x)3 + 43 = (6x + 4)(36x2 - 24x +16)
(6x)3 + 43 = 8(3x + 2)(9x2 - 6x + 4)
Answer: The factor of 216x3 + 64 is 2(3x + 2)(9x2 - 6x + 4).
Example 2: Find the factor of 8x3 + 125y3.
To find: Factor of 8x3 + 125y3, using the sum of cubes formula.
8x3 + 125y3 = (2x)3 + (5y)3
Using the sum of cubes formula,
a3+b3 = (a + b)(a2 - ab + b2)
Put the values,
(2x)3 + (5y)3 = (2x + 5y)((2x)2 – (2x)(5y) + (5y)2)
(2x)3 + (5y)3 = (2x + 5y)(4x2 – 10xy + 25y2)
Answer: The factor of 8x3 + 125y3 is (2x + 5y)(4x2 – 10xy + 25y2).
Example 3: Simplify 193 + 203 using the sum of cubes formula.
Solution: To find 193 + 203
Let us assume a = 19 and b = 20
Using sum of cubes formula a3 + b3 = (a + b) (a2 - ab + b2)
We will substitute these in the a3 + b3 formula
a3 + b3 = (a + b) (a2 - ab + b2)
193+203 = (19+20)(192 - (19)(20)+202)
= (39)(361-380+400)
= (39)(381)
= 14,859
Answer: 193 + 203 = 14859.
FAQ's on Sum of Cubes Formula
What Is the Expansion of Sum of Cubes Formula?
a3 + b3 formula is known as the sum of cubes formula it is read as a cube plus b cube. Its expansion is expressed as a3 + b3 = (a + b) (a2 - ab + b2).
What Is the Sum of Cubes Formula in Algebra?
The sum of cubes formula is one of the important algebraic identity. It is represented by a3 + b3 and is read as a cube plus b cube. The sum of cubes (a3 + b3) formula is expressed as a3 + b3 = (a + b) (a2 - ab + b2).
How To Simplify Numbers Using the Sum of Cubes Formula?
Let us understand the use of the sum of cubes formula i.e., a3 + b3 formula with the help of the following example.
Example: Find the value of 1003 + 23 using the sum of cubes formula.
To find: 1003 + 23
Let us assume that a = 100 and b = 2.
We will substitute these in the formula of the sum of cubes formula that is, a3 + b3
a3 + b3 = (a + b) (a2 - ab + b2)
1003+23 = (100+2)(1002 - (100)(2)+22)
= (102) (10000-200+4)
= (102)(9804)
= 1000008
Answer: 1003 + 23 = 1000008.
How To Use the Sum of Cubes Formula Give Steps?
The following steps are followed while using the sum of cubes formula.
- Firstly observe the pattern of the two numbers whether the numbers have ^3 as power or not.
- Write down the sum of cubes formula of a3 + b3 = (a + b) (a2 - ab + b2)
- substitute the values of a and b in the sum of cubes (a3 + b3) formula and simplify.
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