Question

Factor the sum or difference of two cubes. $$ x^{3}+27 $$

Short answer

\( (x + 3)(x^{2} - 3x + 9) \)

Step-by-step solution

- Identify the cube terms

Recognize that the given expression is a sum of two cubes. The formula for factoring a sum of cubes is: \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2}) \) Here, \( x^{3} + 27 \), where \( x^{3} \) is \( a^{3} \) and \( 27 \) is \( b^{3} \). Identify the cube roots of these terms: \( a = x \) and \( b = 3 \).

- Apply the sum of cubes formula

Using the cube roots identified, plug them into the sum of cubes formula: \( a + b = x + 3 \) and \( a^{2} - ab + b^{2} \) becomes: \(x^{2} - x \cdot 3 + 3^{2} \).

- Simplify the expression

Now, simplify the second factor: \( x^{2} - 3x + 9 \). So the factored form of \( x^{3} + 27 \) becomes: \((x + 3)(x^{2} - 3x + 9) \).

Key concepts

sum of cubes

When we talk about the 'sum of cubes,' we refer to an expression of the form \(a^3 + b^3\). This is a special form in algebra, which can be factored using a specific formula. The sum of cubes formula is:
\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
This formula helps to break down the expression into a product of a binomial and a trinomial, which are both easier to handle.

For example, let's apply this to \(x^3 + 27\):
- First, identify the cube roots: \(a = x\) and \(b = 3\).
- Then, plug these values into the sum of cubes formula:
\(x^3 + 27 = (x + 3)(x^2 - 3x + 9)\).

This transformation makes the equation simpler to work with and better for factoring. Understanding the sum of cubes is a powerful tool in algebra.

factoring polynomials

Factoring polynomials is a key skill in algebra that involves breaking down a polynomial into simpler components, or 'factors,' that when multiplied together give back the original polynomial. This can make solving equations and simplifying expressions much easier.

There are several techniques for factoring polynomials:
- **Common Factor:** Take out the greatest common factor from all terms.
- **Grouping:** Group terms to factor by pairs.
- **Special Forms:** Factor using identities like the difference of squares or the sum/difference of cubes.

In our example of \(x^3 + 27\), we use the 'sum of cubes' formula to factor it, turning it into \((x + 3)(x^2 - 3x + 9)\). This makes it easier to handle compared to the original polynomial. Factoring simplifies complex polynomials and is essential for solving higher-degree equations.

algebraic identities

Algebraic identities are equations that hold true for all values of the variables within them. They serve as shortcuts for simplifying expressions and solving equations. Common algebraic identities include:
- **Difference of Squares:** \(a^2 - b^2 = (a + b)(a - b)\)
- **Sum of Cubes:** \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
- **Difference of Cubes:** \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

These identities enable us to factor and simplify expressions quickly.

In our example, the sum of cubes identity helps to factor \(x^3 + 27\) into \((x + 3)(x^2 - 3x + 9)\). Using these identities proficiently is crucial for success in algebra, making complex problems manageable by breaking them down into simpler parts.

cube roots

A cube root of a number is a value that, when multiplied by itself three times (cubed), gives the original number. For example, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\). In algebra, knowing how to find cube roots is important for factoring and simplifying expressions.

To find the cube roots in our expression \(x^3 + 27\), we:
- Identify \(x^3\) whose cube root is \(x\).
- Identify \(27\) whose cube root is \(3\).

These roots allow us to apply the sum of cubes formula, transforming \(x^3 + 27\) into \((x + 3)(x^2 - 3x + 9)\).
Understanding cube roots and their properties is vital for mastering algebra and effectively working with polynomial identities and factorizations.