Question

Factor the sum or difference of cubes.$x^3-27$

Short answer

The factored form of $x^3-27$ is $(x-3)(x^2+3x+9)$

Step-by-step solution

Identify a and b

In this problem, $x$ is $a$ and 3 is $b$. Here $a^3-b^3$ = $x^3-3^3$ = $x^3-27$. So, $a=x$ and $b=3$.

Apply factoring formula

Using the formula for factoring the difference of cubes: $a^3-b^3=(a-b)(a^2+ab+b^2)$, replace $a$ with $x$ and $b$ with 3. After substitution, factoring expression becomes: $(x-3)((x{)}^2+(x)\ast (3)+(3{)}^2)$

Simplify the expression

The step is to simplify the factored expression to its simplest form. Calculate the square of $x$ and 3, multiply $x$ and 3, and then simplify to obtain: $(x-3)(x^2+3x+9)$

Key concepts

Difference of Cubes

When you have an expression like the one in the problem, recognizing it as a difference of cubes is key. A difference of cubes follows the format: $a^3-b^3$. To factor this correctly, you use the special formula:

• $a^3-b^3=(a-b)(a^2+ab+b^2)$

This formula allows you to break down the expression into a product of two simpler factors. To apply this,

• identify what $a$ and $b$ are in your expression. For instance, if you have $x^3-27$, what you're seeing is $x^3-3^3$.
• Here, $a=x$ and $b=3$, since $27$ is $3^3$.

Each part of the formula corresponds to a different element of the expression and helps simplify it greatly.

Polynomials

Polynomials are expressions that consist of variables, coefficients, and the operations of addition, subtraction, and multiplication. They do not include division by a variable. A polynomial is made up of terms, which are parts of the expression separated by plus or minus signs. In the context of the exercise, the initial expression $x^3-27$ is a polynomial of degree 3. This term 'degree' refers to the highest power of the variable in the polynomial. When factoring a polynomial like the difference of cubes, you're turning one polynomial expression into the product of two or more polynomial expressions. This both simplifies the original expression and makes it easier to work with. In specific math problems or equations, recognizing the degree and format of a polynomial helps you choose the right factoring or simplification method.

Algebraic Expressions

An algebraic expression is a combination of variables and numbers using operations like addition, subtraction, multiplication, and division. Unlike equations, they do not have an equals sign. In our exercise, expressions like $x^3-27$ are known as algebraic expressions. These expressions can be simplified or factored to reveal the underlying structure or to solve equations more easily. When you factor an algebraic expression, you are essentially "breaking it down" into simpler pieces that maintain the original expression's value.Understanding algebraic expressions and the techniques to simplify or factor them, such as using formulas for difference of cubes, helps tremendously in algebra.

• It enables finding solutions to equations.
• It aids in understanding the relationships between different parts of an expression.
• It is a critical skill in any further mathematical study or application.