Problem 29 - Elementary and Intermediate Algebra

Question

Factor. $$ 36 m^{2}+60 m n+25 n^{2} $$

Step-by-Step Solution

Answer

The expression factors to \((6m + 5n)^2\).

Step 1: Identify the Quadratic Form

The expression given is a quadratic trinomial in the form of \(ax^2 + bx + c\). In this case, identify \(a = 36\), \(b = 60\), and \(c = 25\), where the expression is \(36m^2 + 60mn + 25n^2\).

Step 2: Check if it's a Perfect Square Trinomial

A perfect square trinomial takes the form \(a^2 + 2ab + b^2\). Check if \(36m^2 = (6m)^2\), \(25n^2 = (5n)^2\), and \(60mn = 2 \times 6m \times 5n\). All conditions are satisfied, indicating that the trinomial is indeed a perfect square.

Step 3: Write the Perfect Square

Since \(36m^2 + 60mn + 25n^2\) is a perfect square trinomial, it can be factored as \((6m + 5n)^2\). Use the relation \((a + b)^2 = a^2 + 2ab + b^2\) to confirm the factorization.

Key Concepts

Quadratic Trinomials

A quadratic trinomial is an algebraic expression that can be written in the form \(ax^2 + bx + c\). This is a fundamental structure in algebra where we deal with three terms: the quadratic term, the linear term, and the constant term.

• The quadratic term is \(ax^2\), where \(a\) is a constant, and \(x\) represents the variable squared.
• The linear term is \(bx\), where \(b\) is also a constant, representing a linear relationship because \(x\) is to the first power.
• The constant term \(c\) is a simple number without a variable attached to it.

Quadratic trinomials appear quite often in algebra because they can represent real-world problems where relationships are nonlinear. Recognizing a quadratic trinomial is the first step in simplifying expressions or solving equations involving these terms.
In the provided exercise, the quadratic trinomial is \(36m^2 + 60mn + 25n^2\), where:- \(a = 36\)- \(b = 60\)- \(c = 25\)Recognizing these components helps apply specific strategies for further simplification, such as factoring.

Perfect Square Trinomials

A perfect square trinomial is a specific kind of quadratic trinomial where the expression equals the square of a binomial. The general form of a perfect square trinomial is \(a^2 + 2ab + b^2\). Identifying such trinomials allows us to factor them efficiently.
There are some key characteristics:

• The first term, \(a^2\), must be a perfect square.
• The last term, \(b^2\), should also be a perfect square.
• The middle term should be twice the product of \(a\) and \(b\), or \(2ab\).

In the exercise, the trinomial \(36m^2 + 60mn + 25n^2\) is examined:- \(36m^2 = (6m)^2\)- \(25n^2 = (5n)^2\)- \(60mn = 2 \times 6m \times 5n\)
Since all these conditions match, the trinomial can be expressed as a perfect square:
\((6m + 5n)^2\). This simplifies the process of dealing with trinomials, as transformations become more straightforward.

Algebraic Factorization

Algebraic factorization is a powerful technique used to simplify expressions and solve equations. It involves expressing an expression as a product of its factors. In algebra, particularly with trinomials, factoring helps in reducing complexity and revealing the simplest form of expressions.

The steps to factor by recognizing perfect square trinomials are:

• Identify whether the trinomial is a perfect square trinomial.
• Confirm the conditions: ensure both end terms are perfect squares and the middle term aligns with \(2ab\).
• Express the trinomial in factored form, usually as \((a + b)^2\).

Applying this to the trinomial \(36m^2 + 60mn + 25n^2\), we can show:- Recognizing it contains perfect squares allows us to factor it as \((6m + 5n)^2\).
Factorization like this is particularly useful as it not only simplifies the expression but also allows for easier manipulation in solving equations or graphing functions. By mastering algebraic factorization, students unlock a valuable tool in both mathematics and real-world problem solving.