Question

Factor the sum or difference of two cubes. $$ x^{3}+125 $$

Short answer

\( x^3 + 125 = (x + 5)(x^2 - 5x + 25) \)

Step-by-step solution

- Identify the form

Recognize that the expression given is in the form of a sum of two cubes. The general form for the sum of two cubes is given by: \[ a^3 + b^3 \]

- Rewrite the expression

Rewrite the given expression to match the form \[ a^3 + b^3 \]. Here, \(a = x\) and \(b = 5\) since \(5^3 = 125\). Therefore, the expression becomes: \[ x^3 + 5^3 \]

- Apply the factoring formula

Use the sum of cubes factoring formula: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]. Plug in \(a = x\) and \(b = 5\): \[ x^3 + 5^3 = (x + 5)(x^2 - x(5) + 5^2) \]

- Simplify the factors

Simplify the expression inside the parentheses: \[ x^3 + 5^3 = (x + 5)(x^2 - 5x + 25) \]

Key concepts

sum of cubes

One essential algebraic concept is understanding how to factor the sum of cubes. A sum of cubes is written in the form \( a^3 + b^3 \). To identify it, look for two terms, each raised to the power of three, being added together. In the expression \( x^3 + 125 \), we can see it's a sum of cubes because \( x^3 \) and \( 125 = 5^3 \) are both cubes. By recognizing this structure, it makes the factoring process much easier. This form will help us apply the appropriate factoring formulas to simplify the polynomial expression.

factoring formulas

Factoring is breaking down a complex expression into simpler parts called factors. For the sum of cubes, we use a specific formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Applying this formula correctly hinges on identifying the values of \(a\) and \(b\) and substituting them into the formula.

Let's look at our example \( x^3 + 125 \). Here, \( x = a \) and \( 125 = 5^3 \), so \( b = 5 \). Plugging these into the sum of cubes formula, we get: \( (x + 5)(x^2 - 5x + 25) \). This result is obtained by substituting \( a \) with \( x \) and \( b \) with \( 5 \) and then simplifying the expressions within the parentheses. By practicing various problems, you will get comfortable using these formulas efficiently.

polynomial expressions

Polynomials are expressions made up of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. In simpler terms, they look like regular algebraic expressions but include multiple terms. For example, \( x^3 + 125 \) is a polynomial. Understanding how to work with polynomials, like recognizing when special factoring formulas (such as the sum of cubes) apply, is fundamental.

When approaching polynomials, always look for patterns or forms you can use. The sum of cubes is one of those patterns. Factoring polynomials is crucial because it makes solving and simplifying equations much easier. By breaking them down into factors, polynomials become less daunting and more manageable.