Which Expression is a Sum of Cubes

Sum of Cubes Formula

The formula to find the addition of two polynomials, athree+ bthree is known as the sum of cubes formula. Let’s learn more than about the sum of cubes formula with a few solved examples. This factoring formula comes in very handy when solving algebraic expressions of diverse types. Memorizing this formula is also easy and tin be done within a matter of minutes. It is very like to the difference in cubes formula as well.

What Is the Sum of Cubes Formula?

In this section, let usa get farther and understand what exactly does information technology 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 kickoff variable
  • b is the 2d variable

Proof of Sum of Cubes Formula

To prove or verify that sum of cubes formula that is, aiii+ b3= (a + b) (aii – ab + b2) nosotros demand to show here LHS = RHS.
LHS term = athree+ bthree

On Solving RHS term nosotros get,
= (a + b) (a2 – ab + bii)
On multiplying the a and b separately with (a2 + ab + b2) nosotros get
= a (aii – ab + b2) + b(a2 – ab + b2)
= a3
– a2b + abii
+ a2b – ab2+ b3

= aiii
– a2b + aiib + 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 observe the cistron of 216x3+ 64.

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To find: Factor of 216x3+ 64, using the sum of cubes formula.

216x3+ 64 = (6x)3 + iv3

Using the sum of cubes formula,

a3+ b3 = (a + b)(atwo– ab + btwo)

Put the values,

(6x)three + 43 = (6x + 4)((6x)2– 6x × 4 + 42)

(6x)3 + 43 = (6x + iv)(36xii– 24x +16)

(6x)3 + 43 = eight(3x + 2)(9xii– 6x + 4)

Answer: The gene of 216xiii + 64 is two(3x + 2)(9x2– 6x + 4).

Example 2:Find the factor of 8xthree + 125y3.

To find: Gene of 8x3 + 125y3, using the sum of cubes formula.

8xiii + 125ythree = (2x)3 + (5y)three

Using the sum of cubes formula,

a3+biii = (a + b)(atwo– ab + b2)

Put the values,

(2x)3 + (5y)3 = (2x + 5y)((2x)ii – (2x)(5y) + (5y)2)

(2x)3 + (5y)3= (2x + 5y)(4x2 – 10xy + 25y2)

Respond: The factor of 8x3 + 125y3 is (2x + 5y)(4xii – 10xy + 25y2).

Example three:
Simplify 193
+ xx3
using the sum of cubes formula.

Solution:
To find 193
+ 203

Permit us presume  a = nineteen and b = 20
Using sum of cubes formula a3+ b3= (a + b) (aii – ab + b2)
We will substitute these in the aiii + bthree formula
a3+ b3= (a + b) (atwo – ab + b2)
19iii+20iii = (19+20)(192 – (19)(20)+20two)
= (39)(361-380+400)
= (39)(381)
= fourteen,859
Answer: xixthree + twenty3 = 14859.

FAQ’southward on Sum of Cubes Formula

What Is the Expansion of Sum of Cubes Formula?

aiii+ b3 formula is known as the sum of cubes formula information technology is read as a cube plus b cube. Its expansion is expressed as athree+ b3= (a + b) (a2 – ab + b2).

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What Is the Sum of Cubes Formula in Algebra?

The sum of cubes formula is ane of the important algebraic identity. Information technology is represented past a3
+ biii and is read as a cube plus b cube. The sum of cubes (a3+ b3) formula is expressed every bit a3+ b3 = (a + b) (a2 – ab + bii).

How To Simplify Numbers Using the Sum of Cubes Formula?

Let us empathise the use of the sum of cubes formula i.due east., a3+ b3 formula with the help of the post-obit example.
Example:
Find the value of 1003 + 23 using the sum of cubes formula.
To find: 1003 + two3

Let us assume that a = 100 and b = 2.
We volition substitute these in the formula of the sum of cubes formula that is, athree + biii
a3+ bthree= (a + b) (a2 – ab + b2)
1003+23 = (100+2)(1002 – (100)(ii)+two2)
= (102) (10000-200+iv)
= (102)(9804)
= 1000008
Respond: 100iii + twothree = 1000008.

How To Use the Sum of Cubes Formula Requite Steps?

The post-obit steps are followed while using the sum of cubes formula.

  • Firstly discover the blueprint of the 2 numbers whether the numbers have ^3 every bit power or not.
  • Write downwardly the sum of cubes formula of athree + biii
    = (a + b) (a2 – ab + b2)
  • substitute the values of a and b in the sum of cubes (athree+ bthree) formula and simplify.

Which Expression is a Sum of Cubes

Source: https://www.cuemath.com/sum-of-cubes-formula/