Contents

Yes, you can factor $x^3y^3+8$ and get $(xy+2)(x^2y^2-2xy+4)$ as the result. Because all of the terms in this expression are perfect cubes, it will be simpler to use one of the predefined formulae for the factorization of similar terms.

In this complete guide, you will learn how to factor the above expression as well as some concepts related to factorization.

## How To Factor $x^3y^3+8$

In this expression, you can see that both the terms are perfect cubes. Therefore, re-write the expression as: $(xy)^3+(2)^3$. Here, you can use the sum of the cube formula, that is:

$a^3+b^3=(a+b)(a^2-ab+b^2)$

In this expression, $a=xy$ and $b=2$. Substitute these definitions in the above formula to get:

$(xy)^3+(2)^3=[(xy)+2][(xy)^2-(xy)(2)+(2)^2]$

Simplify as follows:

$x^3y^3+8=[xy+2][x^2y^2-2xy+4]$

## How To Factor $x^3+y^3$

The factorization of $x^3+y^3$ is much more simple and easier as compared to $x^3y^3+8$. Here, you just need the direct application of the sum in the cube formula. You can see that $a$ is replaced by $x$ and $b$ is replaced by $y$ in the given expression. Also, it is understood that both $x$ and $y$ are the perfect cubes. Let’s find out the result and see what will be the final form when $a$ will be replaced by $x$ and $b$ will be replaced by $y$.

The sum in cubes formula is $a^3+b^3=(a+b)(a^2-ab+b^2)$. Accordingly, $x^3+y^3=(x+y)(x^2-xy+y^2)$. You can see that these formulae made the calculations and simplifications much easier. It is beneficial to use such formulae when solving an expression containing higher powers of a variable or more than $3$ or $4$ terms.

To make sure that you have applied the correct formula, simply multiply the expression on the right-hand side again. You can see that you will get the expression $x^3+y^3$ back after simplification.

## What Is Factorization?

Factorization or factoring is classified in mathematics as the splitting or breaking of an entity such as a matrix, a polynomial, or a number into a product of some other factors or entities, which when multiplied together give the original polynomial, number, or matrix.

### More Information

Factorization is simply dividing a polynomial or integer into factors that, when multiplied together, yield the existing or initial polynomial or integer.

We use the factorization technique to simplify any quadratic or algebraic equation by representing it as the product of factors rather than expanding the brackets. A variable, an integer, or an algebraic expression can be the factors of any given equation.

### What Is a Polynomial?

Polynomials are algebraic expressions with coefficients or variables. Variables are also referred to as indeterminates. It is not possible to divide a polynomial by a variable. However, you can perform arithmetic operations, namely, multiplication, subtraction, addition, and positive integer exponents for polynomial expressions as well.

### Factorizing Polynomials

A polynomial is an expression that uses an addition or subtraction symbol to separate a mixture of a constant and a variable. Factoring polynomials is the inverse process of multiplying polynomial factors.

Factors of polynomials are zeros of polynomials written in the form of some other linear polynomial. If you divide a polynomial by any one of its factors upon factorization, you will get the remainder of zero.

### What Is a Perfect Cube?

A perfect cube of a number refers to taking the product of a number with itself three times. For instance, $a=b^3$ if $a$ is the perfect cube of $b$. As a result, taking the cube root of a perfect cube yields a natural number rather than a fraction, thus $\sqrt[3]{a}=b$ since it is well known that $64$ is a perfect cube because $\sqrt[3]{64}=4$.

## What Are the Different Types of Factoring Polynomials?

The Grouping method, the Greatest common factor (abbreviated as GCF), the sum or difference in cubes, and the difference in two squares are the four primary types of factoring.

### Greatest Common Factor

To factorize a polynomial, we must first determine its greatest common factor. This method is nothing more than a sort of distributive law reverse process, for example, $x( y + z) = xy +xz$. However, in the case of factorization, it is simply an inverse process: $xy + xz = x(y + z)$, where $x$ can be regarded as the greatest common factor.

#### Example

Factorize the expression $x^2+xy$. In this expression, the greatest common factor is $x$ and can be taken out as $x(x+y)$.

### Factor by Grouping

This technique is also referred to as pair factoring. To find the zeros, a polynomial is grouped in pairs or distributed in pairs.

#### Example

Consider an equation $x^2-x-6$. Now, find out two numbers such that when you add them, the result will be $-1$, and when you multiply them, the result will be $-6$.

Here, $2$ and $-3$ are two numbers such that $2-3=-1$ and $(2)(-3)=-6$. Next, re-write the polynomial as $x^2+2x-3x-6$ or $x(x+2)-3(x+2)$. Now, take $x+2$ as a common factor you will get $(x+2)(x-3)$. Thus, the factors are $(x+2)$ and $(x-3)$.

### Factoring the Sum or Difference in Cubes

The sum or difference of two cubes can be factored into a product of binomial times a trinomial, such as $a^3\pm b^3=(a\pm b)(a^2\pm ab+b^2)$.

#### Example

Take $a=x$ and $b=3$. So the sum of the cubes will be:

$(x)^3+(3)^3=(x+3)(x^2-3x+3^2)$ or $x^3+27=(x+3)(x^2-3x+9)$.

Similarly, $(x)^3-(3)^3=(x-3)(x^2+3x+3^2)$ or $x^3-27=(x-3)(x^2+3x+9)$.

### The Difference in Two Squares

The following formula can be used to factor any polynomial that corresponds to a difference of squares:

$(a^2-b^2)=(a+b)(a-b)$

## Conclusion

This article has been a good source of information on the factorization of $x^3y^3+8$ as well as the concepts relating to factorization, so we have summarized the whole study to gain a better understanding of the concepts presented:

**The factorized form of $x^3y^3+8$ is $(xy+2)(x^2y^2-2xy+4)$.****Factorization or factoring is defined as the breaking or splitting of an entity.****Polynomials are algebraic expressions that consist of variables and coefficients.****A perfect cube of a number refers to taking the product of a number with itself three times.****There are four main types of factoring.**

The easiest way to factor $x^3y^3+8$ is to use one of the common types of factoring, that is “factoring by the sum and difference in cubes.” How about taking the polynomials with more than three terms to have a better command of factoring? This will make you an expert in using various methods for factoring the given expression.