Factor The Sum Of Cubes

Sum of Cubes Formula

The formula to find the improver of two polynomials, a³+ b³ is known as the sum of cubes formula. Permit'due south 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 besides like shooting fish in a barrel and tin exist done within a matter of minutes. It is very like to the deviation in cubes formula besides.

What Is the Sum of Cubes Formula?

In this section, let us go further and sympathise 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)(aⁱⁱ- ab + b²)

where,

a is the kickoff variable
b is the second variable

Sum of cubes formulas

Proof of Sum of Cubes Formula

To bear witness 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 (aⁱⁱ + ab + b²) we get
= a (a^two - ab + b²) + b(aⁱⁱ - ab + bⁱⁱ)
= a³ - aⁱⁱb + ab² + a²b - ab²+ b^three
= a^three - a²b + aⁱⁱb + ab²- ab²+ b³
= a^three - 0 + 0 + bⁱⁱⁱ
= aⁱⁱⁱ + bⁱⁱⁱ
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 216xⁱⁱⁱ+ 64.

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

216x³+ 64 = (6x)^three + 4³

Using the sum of cubes formula,

a³+ b³ = (a + b)(a²- ab + b²)

Put the values,

(6x)ⁱⁱⁱ + iv^three = (6x + iv)((6x)²- 6x × 4 + 4^two)

(6x)³ + four³ = (6x + 4)(36x²- 24x +16)

(6x)³ + 4³ = eight(3x + 2)(9xⁱⁱ- 6x + four)

Answer: The gene of 216x³ + 64 is 2(3x + 2)(9x^two- 6x + iv).

Example 2:Find the cistron of 8x³ + 125y³.

To find: Factor of 8xⁱⁱⁱ + 125y³, using the sum of cubes formula.

8x³ + 125y³ = (2x)³ + (5y)³

Using the sum of cubes formula,

a³+bⁱⁱⁱ = (a + b)(a²- ab + bⁱⁱ)

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 8xⁱⁱⁱ + 125y³ is (2x + 5y)(4xⁱⁱ – 10xy + 25y²).

Instance 3: Simplify 19^three + 20³ using the sum of cubes formula.

Solution: To find nineteen³ + 20³

Let us assume a = 19 and b = 20
Using sum of cubes formula a³+ bⁱⁱⁱ= (a + b) (a² - ab + b²)
We volition substitute these in the aⁱⁱⁱ + b³ formula
a³+ b³= (a + b) (a² - ab + bⁱⁱ)
19³+xx³ = (19+twenty)(xix² - (xix)(20)+20²)
= (39)(361-380+400)
= (39)(381)
= 14,859
Answer: 19ⁱⁱⁱ + 20ⁱⁱⁱ = 14859.

FAQ'due south on Sum of Cubes Formula

What Is the Expansion of Sum of Cubes Formula?

a³+ bⁱⁱⁱ formula is known as the sum of cubes formula it is read equally a cube plus b cube. Its expansion is expressed as aⁱⁱⁱ+ 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. Information technology is represented by a^three + bⁱⁱⁱ and is read every bit a cube plus b cube. The sum of cubes (a³+ b³) formula is expressed as a^three+ b³ = (a + b) (a² - ab + b^two).

How To Simplify Numbers Using the Sum of Cubes Formula?

Allow u.s. sympathise the employ of the sum of cubes formula i.e., a³+ b³ formula with the assist of the following case.
Case: Find the value of 100ⁱⁱⁱ + 2³ using the sum of cubes formula.
To find: 100³ + 2³
Let us presume that a = 100 and b = two.
We volition substitute these in the formula of the sum of cubes formula that is, a^three + b³
a³+ b^three= (a + b) (a² - ab + b^two)
100³+2³ = (100+2)(100² - (100)(2)+2ⁱⁱ)
= (102) (10000-200+four)
= (102)(9804)
= 1000008
Answer: 100ⁱⁱⁱ + 2ⁱⁱⁱ = 1000008.

How To Use the Sum of Cubes Formula Give Steps?

The following steps are followed while using the sum of cubes formula.

Firstly notice the pattern of the two numbers whether the numbers have ^3 as ability or not.
Write downwards the sum of cubes formula of aⁱⁱⁱ + b³= (a + b) (a² - ab + b²)
substitute the values of a and b in the sum of cubes (a³+ b³) formula and simplify.