Problem 39 - A Graphical Approach to College Algebra

Question

Factor each perfect square trinomial completely. $$(2 p+q)^{2}-10(2 p+q)+25$$

Step-by-Step Solution

Answer

\((2p + q - 5)^2\) is the complete factorization.

Step 1: Identify the Structure

The given expression is \( (2p + q)^2 - 10(2p + q) + 25 \). This is in the form of a perfect square trinomial, which typically looks like \( a^2 - 2ab + b^2 \) and factors into \((a-b)^2\).

Step 2: Rewrite as a Single Variable

Notice that the expression can be simplified by letting \( x = 2p + q \). Substitute into the expression to get \( x^2 - 10x + 25 \).

Step 3: Recognize a Perfect Square Trinomial

The expression \( x^2 - 10x + 25 \) is a perfect square trinomial because it matches the pattern \( a^2 - 2ab + b^2 \), where \( a = x \) and \( b = 5 \).

Step 4: Factor the Perfect Square Trinomial

Since \( x^2 - 10x + 25 = (x-5)^2 \), factor the trinomial as \( (x - 5)^2 \).

Step 5: Substitute back the Original Variable

Replace \( x \) with \( 2p + q \). Thus, the factored expression of the original problem is \( ((2p+q) - 5)^2 \).

Step 6: Simplify the Factored Expression

The expression simplifies to \((2p + q - 5)^2\).

Key Concepts

Algebra

Algebra is the foundation for solving expressions and equations by finding unknown values or simplifying complex mathematical statements. When you encounter expressions like

• \((2p + q)^2 - 10(2p + q) + 25\)
• it involves identifying underlying structures

In this exercise, we are dealing with a perfect square trinomial. This type of polynomial represents a special pattern that can be simplified using algebraic techniques. We use substitutions, simplifications, and recognition of patterns to break down the expressions into their familiar forms.
Substituting variables, like letting \(x = 2p + q\), is an important technique in algebra. It allows you to handle simpler versions of complicated expressions and find solutions more easily. Once the expression is simplified, re-substitution is used to return to the original variables, retaining the correct relationship between the terms.

Polynomials

Polynomials are expressions that consist of variables and coefficients, combined using operations like addition, subtraction, and multiplication. Specifically, perfect square trinomials have a special form

• \(a^2 - 2ab + b^2\)
• which factors into \((a-b)^2\).

The given expression \((2p + q)^2 - 10(2p + q) + 25\) is a polynomial where recognizing the structure as a trinomial helps to factor it easily. After simplifying by substitution, the trinomial becomes \(x^2 - 10x + 25\).
In examining the polynomial structure, if we identify \(x^2 - 10x + 25\) as a perfect square, it tells us that there's a pattern of squared terms surrounding a linear term. This pattern enables the equation to be expressed in its factored form as \((x - 5)^2\). Understanding these nuances in polynomial structures is key for solving and interpreting algebraic expressions.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree 2, generally in the form \(ax^2 + bx + c\). When these take the form of a perfect square trinomial, like \(x^2 - 10x + 25\), they can be rewritten for simplicity.
Identifying a quadratic expression as a perfect square trinomial allows for straightforward factoring. In our example, by recognizing the pattern \(a^2 - 2ab + b^2\), we deduced that \(a = x\) and \(b = 5\). Since \((x-5)^2 = x^2 - 10x + 25\), this indicates that when squared, the simplified quadratic expression generates the original trinomial.

This approach is integral in simplifying quadratic expressions and solving equations efficiently, as it reduces complex expressions into their simplest squares, managing the structure and making calculation more systematic for consistent results.