Problem 19 - Elementary and Intermediate Algebra

Question

Factor. $$ a^{3}+64 $$

Step-by-Step Solution

Answer

The expression factors to \((a + 4)(a^2 - 4a + 16)\).

Step 1: Identify the Sum of Cubes

The expression given is in the form of a sum of cubes, specifically \(a^3 + 4^3\). The numbers 64 and 4 correspond to the cube and its base, respectively. This can be factored using the sum of cubes formula: \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\).

Step 2: Apply the Sum of Cubes Formula

Substitute \(a\) for \(x\) and \(4\) for \(y\) in the sum of cubes formula. This gives us:\[a^3 + 4^3 = (a + 4)(a^2 - 4a + 16)\]

Step 3: Verify the Factorization

Expand the factored expression to ensure it equals the original expression. Multiply the terms:- First, \((a + 4)\) gives two products: \(a(a^2 - 4a + 16)\) and \(4(a^2 - 4a + 16)\).- Calculate \(a imes (a^2 - 4a + 16) = a^3 - 4a^2 + 16a\).- Calculate \(4 imes (a^2 - 4a + 16) = 4a^2 - 16a + 64\).- Combine to get \(a^3 + 0a^2 + 0a + 64\).Thus confirming \(a^3 + 64\) is equal to \((a + 4)(a^2 - 4a + 16)\).

Key Concepts

Sum of Cubes

When dealing with the sum of cubes, it's essential to recognize the form it takes: \(x^3 + y^3\). This is a special kind of polynomial that can be factored using a specific formula. The sum of cubes formula is:

• \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)

This formula helps to break down and simplify polynomials that might initially seem complex. In the given exercise, \(a^3 + 64\), the number 64 is actually \(4^3\), making it a perfect candidate for applying the sum of cubes formula. By identifying \(a\) and \(4\) as our \(x\) and \(y\), respectively, the expression \(a^3 + 64\) can be factored into \((a + 4)(a^2 - 4a + 16)\).

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. They form the backbone of algebra and are used to convey relationships, operations, and equate values. In our expression \(a^3 + 64\), both \(a^3\) and 64 are parts of the overall expression. Understanding the components:

• \(a^3\) is a term where \(a\) is raised to the power of 3, representing \(a\times a\times a\).
• 64 is a numerical term representing a constant that does not change.

In algebraic expressions, terms are combined using operations such as addition, subtraction, multiplication, and division. Recognizing these terms and operations is crucial when applying formulas and solving equations.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of polynomials with simpler expressions. It's like dismantling a large puzzle into smaller, more manageable pieces. This process not only simplifies the expression but also provides insights into its structure and roots. For \(a^3 + 64\), we identified it as a sum of cubes:

• The expression can be factored using the formula for the sum of cubes, yielding \((a + 4)(a^2 - 4a + 16)\).

After factorization, you are left with a linear and a quadratic polynomial, often simpler to analyze or solve. Let's briefly verify the factorization by expanding it back:

• Multiply \((a + 4)\) with each term in \((a^2 - 4a + 16)\).
• You'll arrive back at \(a^3 + 64\), confirming correct factorization.

Factorization is a useful tool in algebra for simplifying expressions and solving equations more efficiently.