Problem 105 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x y+c y^{2}\). \(m^{2}-64 m n-65 n^{2}\)

Step-by-Step Solution

Answer

(m - 65n)(m + n)

Step 1 - Identify coefficients

Identify the coefficients of the trinomial in the form \(x^{2} + bxy + c y^{2}\). In the trinomial \(m^{2} - 64mn - 65n^{2}\), the coefficients are: \(a = 1\), \(b = -64\), and \(c = -65\).

Step 2 - Find two numbers that multiply to \(ac\) and add to \(b\)

Calculate \(ac = (1)(-65) = -65\). Now, find two numbers that multiply to -65 and add to -64. These numbers are -65 and 1 because \(-65 \times 1 = -65\) and \(-65 + 1 = -64\).

Step 3 - Rewrite the middle term using the found numbers

Rewrite the trinomial by splitting the middle term using -65 and 1. The trinomial \(m^{2} - 64mn - 65n^{2}\) becomes \(m^{2} - 65mn + mn - 65n^{2}\).

Step 4 - Factor by grouping

Factor by grouping the terms in pairs: \((m^{2} - 65mn) + (mn - 65n^{2})\). Factor out the greatest common factor from each pair: \(m(m - 65n) + n(m - 65n)\).

Step 5 - Factor out the common binomial

Factor out the common binomial \((m - 65n)\): \((m - 65n)(m + n)\).

Key Concepts

Trinomial Factoring

Trinomial factoring is a key skill in algebra. It involves expressing a trinomial, which is a polynomial with three terms, as a product of two binomials. This process can simplify solving equations and understanding polynomial expressions.

For example, consider the trinomial given in the problem: \(m^{2} - 64mn - 65n^{2}\). The goal is to rewrite this trinomial as a product of two binomials.

The steps involve:

• Identifying the coefficients of each term
• Finding two numbers that multiply to the product of the first and last coefficients but add to the middle coefficient
• Splitting the middle term and grouping terms
• Factoring out common factors from each group
• Factoring out the common binomial

By carefully following these steps, we simplify the trinomial factoring process.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition, subtraction, multiplication). Understanding how to manipulate them is vital in solving many mathematics problems.

In the given problem, we start with the expression: \(m^{2} - 64mn - 65n^{2}\). Here, the expressions include variables \(m\) and \(n\), and coefficients like \(-64\) and \(-65\).

The steps to manage an algebraic expression involve:

• Identifying coefficients and variable parts
• Using operations to combine or simplify terms
• Applying mathematical rules, such as factoring

By practicing these steps, you become more comfortable with handling algebraic expressions in various contexts.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler components, which when multiplied together yield the original polynomial. This is useful for solving equations, simplifying expressions, and other algebraic operations.

For the polynomial \(m^{2} - 64mn - 65n^{2}\), we followed a method to factor it into \((m - 65n)(m + n)\).

The general process includes:

• Identifying the polynomial type (trinomial in this case)
• Finding pairs of numbers that satisfy specific multiplication and addition properties
• Rewriting and grouping terms
• Extracting common factors

For our trinomial:

• Identify coefficients: \(a = 1, b = -64, c = -65\)
• Calculate \(ac = -65\), find numbers \(-65\) and \(1\)
• Rewrite trinomial: \(m^{2} - 65mn + mn - 65n^{2}\)
• Group terms: \((m^{2} - 65mn) + (mn - 65n^{2})\)
• Factor: \(m(m - 65n) + n(m - 65n) = (m - 65n)(m + n)\)

Successful factorization makes polynomial math more manageable and serves as a foundational skill in algebra.