Problem 30 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ y^{5}+6 y^{4} $$

Step-by-Step Solution

Answer

The GCF is \( y^4 \); factorization: \( y^4 (y + 6) \).

Step 1: Identify the Greatest Common Factor (GCF)

To factor out the GCF, we need to identify the greatest common factor in the terms of the polynomial. The given polynomial is \( y^5 + 6y^4 \). Both terms contain the factor \( y^4 \), which is the highest power of \( y \) that is present in both terms.

Step 2: Factor Out the GCF

Once the GCF is identified, we factor it out of each term in the polynomial. The GCF here is \( y^4 \), so we factor \( y^4 \) out of each term of the polynomial: \( y^5 + 6y^4 = y^4(y^{5-4} + 6) = y^4(y + 6) \).

Step 3: Verify the Factored Expression

To verify the solution, distribute the factored term back to ensure it equals the original polynomial. Distributing gives \( y^4(y + 6) = y^5 + 6y^4 \). Since this is the original polynomial, the factorization is correct.

Key Concepts

Factoring

Factoring a polynomial involves expressing it as a product of simpler expressions. This makes it easier to understand and solve equations involving the polynomial. In the exercise "\( y^5 + 6y^4 \)", factoring transforms the complex polynomial into a simpler form without changing its value.

When we factor a polynomial, we are essentially dividing it to see what elements (or parts) are common and can be easily multiplied together to reassemble the original expression.

- Factoring simplifies equations, making them easier to solve.- It breaks down complicated polynomials into easy-to-manage components.- Provides a clearer view of the polynomial's roots or solutions.

In practice, we start by identifying components common to all terms in the polynomial, known as factors. Once identified, these factors are extracted or 'factored out,' simplifying the polynomial expression.

Greatest Common Factor

The Greatest Common Factor (GCF) is crucial in the process of factoring polynomials. It is the largest expression that can evenly divide each term in a polynomial, allowing you to "factor it out" and simplify the polynomial as in our example, "\( y^5 + 6y^4 \)."

• Identifying the GCF: Look for the highest power of each variable shared among all terms. In the exercise, both terms have the factor "\( y^4 \)" in common, making it the GCF.
• Using the GCF: Once the GCF is identified, it can be factored out, which simplifies the expression. For the polynomial given, factoring out the GCF \( y^4 \) gives \( y^4(y + 6) \).
• Verification: To ensure the factoring was done correctly, multiply the factored terms back out. This should give the original polynomial if done correctly.

Remember, finding the GCF helps minimize the complexity of the polynomial which is particularly beneficial when solving equations or integrating polynomials into larger expressions.

Polynomial Expressions

Polynomial expressions are sums or differences of terms where each term is a product of a number (coefficient) and a variable raised to a non-negative integer exponent. They are widely used in algebra and higher-level mathematics.

Consider the polynomial "\( y^5 + 6y^4 \)":

• Structure: It consists of two terms: \( y^5 \) and \( 6y^4 \).
• Terms: Each term is composed of a coefficient (1 and 6 respectively) and a variable base \( y \) raised to an exponent.
• Operations: You can perform operations such as addition, subtraction, and multiplication on these terms.

Polynomials are flexible and applicable to numerous scenarios, including graphing equations, solving quadratic equations, and modeling real-world situations in science and engineering.
Understanding how to manipulate them, such as factoring using the GCF, is fundamental in algebraic problem solving.