Which Expression is a Sum of Cubes

Sum of Cubes Formula

The formula to find the addition of two polynomials, a^three+ b^three 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 = a^three+ b^three

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

Examples on Sum of Cubes Formula

Example1: Use the sum of cubes formula to observe the cistron of 216x³+ 64.

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)(a^two– ab + b^two)

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 8x^three + 125y³.

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

8xⁱⁱⁱ + 125y^three = (2x)³ + (5y)^three

Using the sum of cubes formula,

a³+bⁱⁱⁱ = (a + b)(a^two– 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 + b^three formula
a³+ b³= (a + b) (a^two – ab + b²)
19ⁱⁱⁱ+20ⁱⁱⁱ = (19+20)(19² – (19)(20)+20^two)
= (39)(361-380+400)
= (39)(381)
= fourteen,859
Answer: xix^three + twenty³ = 14859.