Problem 45 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. $$ 15 p^{4}+31 p^{3} q+2 p^{2} q^{2} $$

Step-by-Step Solution

Answer

The trinomial can be grouped as \(p^3(15p + 31q)\).

Step 1: Identify the Terms

Identify the four terms of the trinomial. Here, the terms are: \(15p^4\), \(31p^3q\), and \(2p^2q^2\). Although the trinomial usually has three terms, notice we can separate it for grouping based on potential factoring opportunities.

Step 2: Group Terms

Group the first two terms and factor out the greatest common factor (GCF) from each group. From \(15p^4 + 31p^3q\), factor out \(p^3\), leaving \(p^3(15p + 31q)\).

Step 3: Factor by Grouping

The second group should be \(0\) as there is no fourth term in our case. Therefore we already have one grouped factor which is \(p^2\).

Step 4: Rearrange and Simplify

Rewrite the expression in its grouped form, using the factor from previous step: \(p^3(15p + 31q)\). Since the expression was initially a trinomial but not easily factorable by grouping, ensure that this form is double-checked.

Key Concepts

Trinomial

A trinomial is a type of polynomial consisting of three distinct terms. In arithmetic and algebra, polynomials are used to describe expressions that include variables, coefficients, and constants. Trinomials are specifically composed of three monomials that are usually connected by addition or subtraction.

Each term in a trinomial has a different degree— the power to which a variable is raised. It is important to identify the degrees because they aid in structuring the equation for simplification and factoring purposes. In the problem we’re considering, however, the expression is modified to fit a grouping strategy.

Understanding trinomials helps in breaking down complex expressions into more manageable parts.

Grouping

Factoring by grouping is a technique often employed when working with polynomials. The method involves rearranging the terms of the polynomial expression in such a way that each group shares a common factor.

To apply this technique, you:

• Identify parts of the polynomial that can be grouped together.
• Factor out the greatest common factor (GCF) from each group.
• Combine the results to simplify the expression further.

In our exercise, we are dealing with a tricky situation where the expression appears to initially have three distinct terms. To apply grouping effectively, we treat the terms cleverly to reveal factors that can be separated and combined for simplification.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is an essential concept in factoring polynomials. It is the largest factor that divides two or more terms without leaving a remainder. Finding the GCF is key to simplifying polynomial expressions through factoring.

In our example, we looked for the GCF within each group of terms:

• For the first group, consisting of the terms \(15p^4\) and \(31p^3q\), the GCF is \(p^3\).
• Factoring \(p^3\) from this group yields \(p^3(15p + 31q)\).

Recognizing and removing the GCF allows us to break down the polynomial and reduce its complexity, making it far easier to manage and solve.

Algebraic Expressions

Algebraic expressions form the cornerstone of algebra, encompassing different terms with variables raised to any power, combined using arithmetic operations. In our case, the polynomial \(15p^4+31p^3q+2p^2q^2\) is an algebraic expression comprising terms with both coefficients and variables.

Dealing with algebraic expressions often requires simplifying and factoring to solve equations, which helps outline solutions to real-world problems. Factoring a polynomial includes examining the terms to identify and extract common factors, facilitating the simplification of the expression.

Understanding algebraic expressions aids in discerning strategies like grouping and applying operations such as finding the GCF, ultimately leading to clear and concise solutions.