Problem 31 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 6 y^{4}+2 y^{3} $$

Step-by-Step Solution

Answer

The GCF is \(2y^3\); the factored polynomial is \(2y^3(3y + 1)\).

Step 1: Identify the Terms

The polynomial given is \(6y^4 + 2y^3\). The terms in this polynomial are \(6y^4\) and \(2y^3\).

Step 2: Determine the GCF of the Coefficients

The coefficients are 6 and 2. The greatest common factor (GCF) of 6 and 2 is 2.

Step 3: Determine the GCF of the Variables

The variable part of the first term is \(y^4\) and for the second term is \(y^3\). The GCF of \(y^4\) and \(y^3\) is the smallest power of \(y\), which is \(y^3\).

Step 4: Combine the GCFs

Combine the GCF of the coefficients and the GCF of the variables: \(2y^3\) is the GCF of the polynomial \(6y^4 + 2y^3\).

Step 5: Factor out the GCF

Divide each term by the GCF \(2y^3\):- \(\frac{6y^4}{2y^3} = 3y\)- \(\frac{2y^3}{2y^3} = 1\)Thus, \(6y^4 + 2y^3 = 2y^3(3y + 1)\).

Key Concepts

Understanding the Greatest Common Factor

The greatest common factor, or GCF, is the largest factor that divides two or more numbers. When dealing with polynomials, finding the GCF is a crucial step in simplifying expressions. The GCF is important to simplify the polynomial by factoring it out.

Finding the GCF involves these steps:

• Identify the coefficients: Examine the numerical part of each polynomial term. For example, in \(6y^4 + 2y^3\), the coefficients are 6 and 2.
• Determine the GCF of the coefficients: For the numbers 6 and 2, determine the largest number that divides both. Here, it is 2.
• Consider variable parts: Look at the variables present. Here, it involves powers of \(y\): \(y^4\) and \(y^3\).
• GCF of variable powers: Choose the smallest power of the common variable. Here, \(y^3\) is the smallest power.

The GCF, \(2y^3\), is used to simplify or "factor out" from the given polynomial, making problems easier to handle.

Identifying Polynomial Terms

A polynomial is made up of terms. A term consists of a coefficient and variables raised to an exponent. For example, in \(6y^4 + 2y^3\), there are two polynomial terms. A good grip on identifying these parts helps in factoring polynomials correctly.

Breakdown of terms includes:

• Naming the terms: Each chunk of the polynomial is called a term. In the example \(6y^4\) and \(2y^3\), both represent individual terms.
• Coefficients and variables: Each term has a numerical part, the coefficient (like 6 and 2), and variables with exponents (like \(y^4\) and \(y^3\)).

Understanding the structure of polynomial terms ensures correct computation when applying factoring techniques.

Using Factoring Techniques

Factoring is a valuable tool in algebra that involves breaking down a polynomial into simpler components, making it easier to solve. The technique involves taking out the GCF to simplify the polynomial.

Steps for factoring polynomials include:

• Locate the GCF: Identify the greatest common factor for coefficients and variables.
• Divide each term by the GCF: For example, divide \(6y^4\) and \(2y^3\) by \(2y^3\). This simplifies to \(3y + 1\).
• Express in factored form: The polynomial \(6y^4 + 2y^3\) becomes \(2y^3(3y + 1)\). This step emphasizes showing the polynomial as a product of its factors.

Mastering these techniques not only simplifies polynomials but also prepares students for solving equations and other advanced math concepts.