Problem 42 - Elementary and Intermediate Algebra

Question

Factor. $$ 8 y^{3}+8 $$

Step-by-Step Solution

Answer

Factor the expression as \( 8(y+1)(y^2-y+1) \).

Step 1: Identify Common Factors

Examine the expression to identify any common factors among the terms. Here, both terms share a factor of 8, so we can factor it out as follows: \[ 8(y^3 + 1) \]

Step 2: Recognize Special Binomial Form

The expression inside the parentheses, \( y^3 + 1 \), is a sum of cubes, since it can be rewritten as \( y^3 + 1^3 \). We will apply the sum of cubes formula, \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \).

Step 3: Apply Sum of Cubes Formula

Set \( a = y \) and \( b = 1 \) to apply the sum of cubes formula: \[ (y + 1)(y^2 - y imes 1 + 1^2) = (y+1)(y^2-y+1) \]

Step 4: Combine Factors Fully

Combine the common factor with the result from the sum of cubes formula. Substitute back into the original factorization to get: \[ 8(y+1)(y^2-y+1) \]

Key Concepts

Understanding the Sum of Cubes

When working with polynomials, recognizing certain patterns can simplify the factoring process. One such pattern is the **sum of cubes**. Specifically, an expression in the form of \( a^3 + b^3 \) can be factorized using the formula \( (a + b)(a^2 - ab + b^2) \). This means that anytime you encounter two cubed terms being added together, you might be dealing with a sum of cubes.In the expression \( y^3 + 1 \), we can rewrite the number 1 as \( 1^3 \) to see the sum of cubes more clearly: \( y^3 + 1^3 \). By setting \( a = y \) and \( b = 1 \), we can apply the formula directly, leading us to \( (y+1)(y^2 - y \times 1 + 1^2) = (y+1)(y^2-y+1) \).By understanding and recognizing the sum of cubes, you can factor these types of expressions effectively, making your work with polynomials simpler and more efficient.

Identifying Common Factors

Before diving into more complicated factoring methods, it's crucial to check for **common factors**. Common factors are numbers or variables that divide each term in the polynomial without leaving a remainder. Factoring out the greatest common factor can simplify the expression significantly and make the problem more manageable.In the example \( 8y^3 + 8 \), both terms share the common factor of 8. By factoring this out, the expression simplifies to \( 8(y^3 + 1) \). This step reduces the complexity of the polynomial and prepares it for further factoring or additional simplification.Recognizing common factors is an essential skill in algebra. It streamlines the process and often reveals other patterns, like binomials or special forms, that can be factored further.

The Process of Polynomial Factorization

**Polynomial factorization** is the process of breaking down a polynomial into the product of simpler polynomials. This makes working with them easier, especially when solving polynomial equations or simplifying expressions.The process starts with checking for common factors, as we saw with the example \( 8(y^3 + 1) \). Once common factors are factored out, the remaining expression is analyzed for special forms or patterns, like the sum of cubes. In our example, using the sum of cubes formula allowed us to factor the expression further into \( 8(y+1)(y^2-y+1) \).Polynomial factorization is a key skill in algebra, essential for simplifying complex expressions and solving equations efficiently. Understanding the steps involved, like identifying common factors and recognizing special binomial forms, unlocks more straightforward and quicker solutions.