Problem 47 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ y\left(x^{2}+2\right)+3\left(x^{2}+2\right) $$

Step-by-Step Solution

Answer

The factored form is \((x^2 + 2)(y + 3)\).

Step 1: Identify Common Factor

First, observe the given polynomial: \( y(x^2 + 2) + 3(x^2 + 2) \). Notice that both terms in the expression share a common factor, which is \( (x^2 + 2) \). This is the Greatest Common Factor (GCF) for the expression.

Step 2: Factor Out the GCF

Now that we have identified \( (x^2 + 2) \) as the common factor, factor it out from each term of the expression. \[y(x^2 + 2) + 3(x^2 + 2) = (x^2 + 2)(y + 3)\]By factoring \( (x^2 + 2) \) from both terms, we are left with \( y \) and \( 3 \) as the other part of each term, respectively.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, often abbreviated as GCF, is a key concept when working with polynomials. It represents the largest expression that divides all terms of a polynomial without leaving a remainder. Identifying the GCF is the first step when trying to simplify or factor a polynomial expression.
When dealing with two or more terms in a polynomial, like in our example expression \( y(x^2 + 2) + 3(x^2 + 2) \), the GCF is the expression common in all terms. Here, \( (x^2 + 2) \) is evident in both parts of the polynomial, making it the GCF.
Understanding how to find the GCF involves:

• Looking for terms or factors that are repeated across the polynomial expressions.
• Recognizing constants, variables, or larger polynomial expressions that are common.

Finding the GCF simplifies calculations and helps break down complex polynomial problems into more manageable parts. It's essential to develop this skill to factor polynomial expressions effectively.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables, coefficients, and exponents. They are constructed from terms, which can be combined using addition, subtraction, multiplication, and exponentiation.
In the polynomial \( y(x^2 + 2) + 3(x^2 + 2) \), we see two terms. Each term is made up of a coefficient and a factor, where \( x^2 + 2 \) is common to both terms. It’s important to understand the structure of a polynomial because this knowledge assists in identifying patterns and applicable factoring techniques.
Key features of polynomial expressions include:

• Degrees: The highest exponent of the variable in a polynomial. In \( x^2 + 2 \), the degree is 2.
• Terms: Individual components of a polynomial added or subtracted together. In this example, \( y(x^2+2) \) and \( 3(x^2+2) \) are terms.
• Coefficients: Numbers that multiply the variable terms. In this case, coefficients are \( y \) and \( 3 \).

Understanding polynomial expressions is crucial because it helps in simplifying expressions and solving equations, especially in identifying common factors and factoring techniques.

Factoring Techniques

Factoring is a technique used to simplify polynomial expressions by breaking them down into products of simpler expressions. Identifying and applying the right factoring technique can make solving polynomial equations much easier.
One common factoring technique is factoring out the Greatest Common Factor (GCF), which we've seen in this exercise. When you factor out the GCF, you divide each term in the polynomial by the GCF and then rewrite the expression as a product of the GCF and the resulting simplified expression. In our example, the polynomial \( y(x^2 + 2) + 3(x^2 + 2) \) was factored into \( (x^2 + 2)(y + 3) \). This shows the product of the GCF \((x^2 + 2)\) and the remaining terms \((y + 3)\).
Some other key factoring techniques include:

• Factoring trinomials: Breaking down a three-term polynomial into the product of two binomials.
• Difference of squares: Recognizing terms in the form \( a^2 - b^2 \) and factoring them as \((a + b)(a - b)\).
• Grouping: Particularly useful for polynomials with four or more terms, grouping involves factoring by subsets of terms.

Mastering these techniques allows for efficient problem-solving and deeper comprehension of polynomial behavior.