The formula to find the addition of two polynomials,a3+ b3is 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 withina 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 tothe 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 ofSum 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 RHStermwe get,
= (a +b) (a2- ab + b2)
On multiplying the a and b separatelywith(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
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Examples on Sum of Cubes Formula
Example1:Use the sum of cubes formula to find the factor of 216x3+ 64.
To find: Factor of216x3+ 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 of8x3+ 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+ 125y3is(2x + 5y)(4x2– 10xy + 25y2).
Example 3: Simplify 193 + 203 using the sum of cubesformula.
Solution: To find193 + 203
Let us assume a = 19 and b = 20
Using sum of cubes formulaa3+ b3= (a + b) (a2- ab + b2)
We will substitute these in thea3+ b3formula
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 CubesFormula
What Is the Expansion of Sum of Cubes Formula?
a3+b3formula is known as the sum of cubes formula it is read as a cube plus b cube. Its expansion is expressed asa3+b3= (a +b) (a2- ab + b2).
What Is theSum of Cubes Formula in Algebra?
The sum of cubesformula is one of the important algebraic identity. It is represented bya3 + b3and is read as a cube plus b cube. The sum of cubes (a3+b3) formula is expressed asa3+b3= (a +b) (a2-ab + b2).
How To Simplify Numbers UsingtheSum of Cubes Formula?
Let us understand the use of the sum of cubes formula i.e., a3+b3formula with the help of the following example.
Example: Find the value of 1003+ 23using the sum of cubesformula.
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 CubesFormula Give Steps?
The following steps are followed while using the sum of cubesformula.
- Firstlyobserve the pattern of the two numbers whether thenumbers have ^3 as power or not.
- Write down the sum of cubes formula ofa3+ b3 = (a +b) (a2-ab + b2)
- substitute the values of a and b in the sum of cubes (a3+b3) formula and simplify.
