Problem 57 - Algebra A Combined Function

Question

Factor. $$ x^{3}+27 $$

Step-by-Step Solution

Answer

The expression \(x^3 + 27\) factors to \((x+3)(x^2 - 3x + 9)\).

Step 1: Recognize the Sum of Cubes

Notice that the expression \(x^3 + 27\) is a sum of cubes. This is because \(x^3\) is the cube of \(x\) and \(27\) is the cube of \(3\). The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).

Step 2: Identify \(a\) and \(b\)

Identify \(a = x\) and \(b = 3\), since \(x^3 = (x)^3\) and \(27 = (3)^3\).

Step 3: Apply the Sum of Cubes Formula

Substitute \(a = x\) and \(b = 3\) into the sum of cubes formula: \((x+3)(x^2 - 3x + 9)\).

Step 4: Verify the Factored Solution

Verify the solution by expanding the factored form to ensure it matches the original polynomial. Expanding \((x+3)(x^2 - 3x + 9)\) should result in \(x^3 + 27\).

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler, non-divisible polynomials, multiplied together. It is like finding the building blocks that can be multiplied to get the original polynomial. In algebra, factorization helps simplify expressions and solve equations.

• It allows for easier manipulation and solving of polynomial equations.
• This technique can reveal roots or zeros of the polynomial, which are values that satisfy the equation when the polynomial is set equal to zero.

In some cases, like with the sum of cubes, factorization requires recognizing special formulas. Applying these formulas can simplify what initially seems complex. Understanding these factorization techniques is crucial, especially for solving higher-degree polynomial equations.

Cubic Expressions

Cubic expressions involve variables that are raised to the third power (or cube). In simpler terms, a cubic expression generally takes the form of \(ax^3 + bx^2 + cx + d\).

• Cubic expressions can represent a wide variety of mathematical situations, from geometrical problems involving volumes to more abstract algebraic problems.
• Recognizing when you're dealing with a cubic expression is important because it determines the types of strategies you'll use to solve or factor the equation.

For example, the expression \(x^3 + 27\) is a simple cubic expression, and here it represents a "sum of cubes." This particular type of cubic expression can be factored using the sum of cubes formula, effectively simplifying the problem into two separate expressions.

Algebraic Formulas

Algebraic formulas are key tools in solving and simplifying algebraic expressions. They provide specific methods for handling certain kinds of algebraic problems, such as factorizing polynomials.

• The sum of cubes formula, \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\), is an excellent example of such a formula.
• This formula is useful because it allows us to see cubic expressions as the sum of two cubes and simplifies the process of factorization.

Using these formulas can save time and reduce complexity in computations. Mastering them often makes a big difference in solving algebraic problems efficiently.