Problem 26 - Beginning and Intermediate Algebra

Question

Divide. $$\frac{m^{2}-6 m-27}{m+3}$$

Step-by-Step Solution

Answer

The simplified expression after dividing \(\frac{m^2-6m-27}{m+3}\) is \(m - 9\).

Step 1: Identify the quadratic equation in the numerator

The numerator is a quadratic equation: \(m^2 - 6m - 27\).

Step 2: Factor the quadratic equation

We will try to find two binomials whose product is equal to the quadratic equation. We need two numbers whose product is -27 and whose sum is -6. Those numbers are -9 and +3. So, the quadratic equation can be factored as \((m - 9)(m + 3)\). Now the expression is: \(\frac{(m - 9)(m + 3)}{m + 3}\) Step 2: Simplify the expression

Step 3: Cancel the common binomial

As both numerator and denominator have a common binomial (m + 3), we can cancel it out. \(\frac{(m - 9)(m + 3)}{m + 3}\) can be simplified to \(m - 9\), as the (m + 3) cancels out. The final simplified expression is: \(m - 9\)

Key Concepts

Factoring Quadratics

Quadratic equations are expressions that can be written in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Factoring quadratics involves rewriting the equation as a product of two binomials. The challenge is to find two numbers that multiply to \( ac \) (the product of \( a \) and \( c \)) and add up to \( b \).In the case of the quadratic equation \( m^2 - 6m - 27 \), we need numbers that multiply to -27 and add up to -6. Here, -9 and +3 satisfy these conditions. Therefore, the quadratic equation can be factored as \( (m - 9)(m + 3) \).Factorizing quadratics makes them easier to work with, especially in operations like division or solving equations. This process helps break down complex expressions into simpler, manageable parts.

Simplifying Expressions

Simplifying expressions is all about reducing them to their most straightforward form. This often involves factoring, canceling out terms, and performing basic arithmetic operations.After factoring the quadratic \( m^2 - 6m - 27 \) into \( (m - 9)(m + 3) \), we simplify by cancelling the common binomial \( (m + 3) \) that appears in both the numerator and the denominator. Hence, \( \frac{(m - 9)(m + 3)}{m + 3} \) simplifies to \( m - 9 \).Simplification not only makes expressions easier to understand but also reveals the core elements of a problem or function. It removes redundancies and highlights the essential parts of the expression.

Common Binomials

Common binomials are terms that appear in both the numerator and the denominator of a fraction. Identifying and canceling them is an essential part of simplifying expressions.In this exercise, the term \( (m + 3) \) is a common binomial present in both the numerator and the denominator after factoring the quadratic expression. By canceling out \( (m + 3) \) in \( \frac{(m - 9)(m + 3)}{m + 3} \), we simplify the expression to \( m - 9 \).The process of canceling common binomials relies on the fundamental fact that any non-zero number divided by itself is 1. Recognizing common binomials can greatly simplify the process of working with algebraic fractions, making it easier to identify the solution.