Problem 14 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. $$ 2 x y^{3}-10 x $$

Step-by-Step Solution

Answer

The polynomial factors to \(2x(y^3 - 5)\).

Step 1: Identify the Greatest Common Factor (GCF)

Start by identifying the common factor in the polynomial. Look at both terms: \(2xy^3\) and \(-10x\). The factors of \(2xy^3\) are \(2, x, y^3\), and the factors of \(-10x\) are \(-10, x\). The common factor between these terms is \(2x\).

Step 2: Factor Out the GCF

Factor out the greatest common factor (GCF), which is \(2x\), from each term in the polynomial. \[2xy^3 - 10x = 2x(y^3 - 5)\]

Step 3: Simplify the Expression

After factoring out \(2x\), you are left with the expression \(y^3 - 5\) inside the parentheses. There are no further common factors, and \(y^3 - 5\) is a simple expression that cannot be factored further using integer factors.

Step 4: Verify Complete Factorization

Confirm that the expression \(2x(y^3 - 5)\) is fully factored. Since \(y^3 - 5\) cannot be further factored using real numbers, the complete factorization is \(2x(y^3 - 5)\).

Key Concepts

Greatest Common Factor

In mathematics, the greatest common factor (GCF) is the largest factor that divides two or more numbers or terms without leaving a remainder. It simplifies expressions and makes calculations easier.
This concept is crucial, especially in polynomial factorization. When working with polynomials, the first step is often to determine the GCF of the terms.

For instance, in the polynomial \(2xy^3 - 10x\), the GCF is found by comparing the factors of each term:

• The term \(2xy^3\) has factors of \(2, x,\) and \(y^3\).
• The term \(-10x\) has factors of \(-10\) and \(x\).

The common factor here is \(x\), and since both terms can also be divided by \(2\), the GCF is \(2x\).

Recognizing the GCF is key to simplifying polynomials. Once identified, it can be factored out by dividing each term of the polynomial by the GCF. This step reduces the polynomial to simpler terms, making it easier to work with in subsequent calculations.

Polynomials

Polynomials are mathematical expressions involving a sum of powers of variables, each multiplied by a coefficient. They can include constants, variables, and exponents, combined with addition, subtraction, and multiplication, but not division by a variable.

In the polynomial \(2xy^3 - 10x\), we have two terms:

• \(2xy^3\), which involves the variable \(y\) raised to the power of 3, along with coefficients \(2\) and \(x\).
• \(-10x\), which is a straightforward term involving the variable \(x\) with a coefficient \(-10\).

Polynomial expressions can vary greatly in complexity. They are crucial in algebra because they form the basis of various functions studied throughout mathematics.

Understanding the structure of polynomials is essential as it helps in identifying appropriate methods for simplification, solving, and manipulation across different mathematical problems.

Factoring Techniques

Factoring is the process of breaking down polynomials into simpler, "factor" polynomials whose product is the original polynomial. It is a critical skill in algebra that helps in solving equations, integrating expressions, and simplifying expressions.

When working on factoring polynomials like \(2xy^3 - 10x\), we follow a systematic approach:1. **Identify the GCF:** As discussed, find the largest common factor that can be factored out from all terms. For \(2xy^3 - 10x\), the GCF is \(2x\).2. **Factor out the GCF:** Rewrite the polynomial by factoring out \(2x\), giving \(2x(y^3 - 5)\). This step simplifies the expression, showing the remaining terms as a product.3. **Check for further factorization:** Investigate the resulting expression. Here, \(y^3 - 5\) does not factor further using integers, so this is the final result.

Remember that these techniques are foundational tools in algebra, providing a method to simplify and solve polynomial equations effectively.