Problem 98 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x y+c y^{2}\). \(m^{2}+6 m n+5 n^{2}\)

Step-by-Step Solution

Answer

\((m + n)(m + 5n)\)

Step 1: Identify coefficients

First, identify the coefficients in the given trinomial. The trinomial is in the form \(m^2 + 6mn + 5n^2\), where:- The coefficient of \(m^2\) is 1.- The coefficient of \(mn\) is 6.- The coefficient of \(n^2\) is 5.

Step 2: Find the pair of numbers

Find two numbers that multiply to 5 (the coefficient of \(n^2\)) and add up to 6 (the coefficient of \(mn\)). These numbers are 1 and 5.

Step 3: Write the factorization

Using the numbers found, write the factorization of the trinomial. The trinomial \(m^2 + 6mn + 5n^2\) factors to: \((m + n)(m + 5n)\)

Key Concepts

Coefficients

Coefficients are the numerical factors in algebraic terms. In the trinomial example given (\(m^2 + 6mn + 5n^2\)), coefficients play a crucial role:

• The coefficient of\(m^2\) is 1. This is because\(1 * m^2 = m^2\).
• The coefficient of\(mn\) is 6. This is the number multiplying the term\(mn\).
• The coefficient of\(n^2\) is 5. This means 5 multiplies the term\(n^2\).

Breaking down these coefficients helps us in the factoring process by identifying the values we need to find pairs for during multiplication and addition.

Factoring Techniques

Factoring trinomials involves identifying numbers that make it easy to split and simplify the expression. For our given trinomial (\(m^2 + 6mn + 5n^2\)), we use the following technique:1. **Identify coefficients**:
This initial step involves pinpointing the coefficients (1, 6, and 5 in our example).
2. **Find pair of numbers**:
Look for two numbers that multiply to the last coefficient (5) and add up to the middle coefficient (6). Here, those numbers are 1 and 5.3. **Write the factorization**:
Establish the simplified factors using these numbers. In this case, the trinomial\(m^2 + 6mn + 5n^2\) factors to:\((m + n)(m + 5n)\).By systematically identifying and using these steps, we can simplify and solve complex algebraic expressions efficiently.

Algebraic Expressions

Algebraic expressions consist of variables and coefficients arranged through operations like addition, subtraction, and multiplication. In our example (\(m^2 + 6mn + 5n^2\)), this trinomial can be simplified using algebraic techniques.
1. **Understand the Structure**:
A trinomial has three terms. Here, the terms are\(m^2\),\(6mn\), and\(5n^2\).2. **Interaction of Terms**:
The terms interact through their coefficients and variables (m and n in our example). To factor the expression, look at these interactions to determine the pairs for multiplication and addition.3. **Simplification**:
Use the identified numbers to rewrite the expression in a simple factored form, resulting in\((m + n)(m + 5n)\).By understanding these components, one can better tackle and simplify algebraic problems, leading to clearer and more accurate solutions.