Problem 53 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{m^{2}+m-20}{m} \div \frac{4-m}{m}\)

Step-by-Step Solution

Answer

The simplified form of the expression is \(-m - 5\).

Step 1: Rewrite the Division as Multiplication

In algebra, when dividing by a fraction, we multiply by its reciprocal. Thus, the problem \( \frac{m^{2}+m-20}{m} \div \frac{4-m}{m} \) becomes \( \frac{m^{2}+m-20}{m} \times \frac{m}{4-m} \).

Step 2: Simplify the Expression

We notice that in the expression \( \frac{m^{2}+m-20}{m} \times \frac{m}{4-m} \), the \( m \) in the numerator of the second fraction can cancel with the \( m \) in the denominator of the first fraction, leaving us with \( \frac{m^{2}+m-20}{4-m} \).

Step 3: Factor the Quadratic Expression

Factor the quadratic expression \( m^{2} + m - 20 \). Look for two numbers that multiply to \(-20\) and add up to \(1\). These numbers are \(5\) and \(-4\). So, we can factor the expression as \((m + 5)(m - 4)\).

Step 4: Substitute and Simplify Further

Substitute the factored form into the expression: \( \frac{(m + 5)(m - 4)}{4 - m} \). Notice that \(m - 4\) and \(4 - m\) are negatives of each other, so \((m - 4) = -(4 - m)\).

Step 5: Simplify the Fraction

Recognize that \((m - 4)/(4 - m) = -1\). Hence, the expression \( \frac{(m + 5)(m - 4)}{4 - m} = (m + 5) \times (-1) = - (m + 5) \). Simplify to \(-m - 5\).

Key Concepts

Factorization

Factorization is a crucial technique in algebra used to break down equations or expressions into products of simpler expressions or polynomials. It helps to simplify problems and discover common factors. In the given original exercise, we are dealing with a quadratic expression:

• The expression is \( m^2 + m - 20 \).
• To factor it, we need to find two numbers that multiply to \(-20\) and add up to \(1\).
• The factors of \(-20\) that fulfill this condition are \(5\) and \(-4\).
• Thus, we can rewrite the quadratic as \((m + 5)(m - 4)\).

Breaking down the quadratic expression is essential in revealing components that can eventually be canceled or simplified, leading to a simpler form of the initial problem.
Factorization helps students recognize patterns in expressions and make complex algebraic problems more manageable.

Reciprocal

The concept of reciprocals is fundamental when dealing with division, especially in algebra. The reciprocal of a number or an expression is essentially switching its numerator and denominator if it is a fraction. In simpler terms:

• For a given fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
• Division by a fraction \( \frac{4 - m}{m} \) becomes multiplication by its reciprocal \( \frac{m}{4 - m} \).

Understanding reciprocals allows us to change division problems into multiplication problems, which are often easier to handle. So, in our exercise:

• We rewrite \( \frac{m^2+m-20}{m} \div \frac{4-m}{m} \) as \( \frac{m^2+m-20}{m} \times \frac{m}{4-m} \).

Using reciprocals simplifies calculations and aids in solving the original algebraic expression more efficiently.

Simplification

Simplification in algebra involves reducing expressions to their simplest form. By eliminating common factors or using algebraic identities, expressions become more accessible to work with. In the exercise:

• First, cancel out similar terms. Here, the \( m \) in the numerator of the second fraction cancels with the \( m \) in the denominator of the first fraction, resulting in \( \frac{m^2 + m - 20}{4 - m} \).
• Next, substitute the factored form from the previous steps, leading to \( \frac{(m + 5)(m - 4)}{4 - m} \).
• Recognize that \( m - 4 \) and \( 4 - m \) are negatives of each other: \( (m - 4) = - (4 - m) \).
• This allows for further simplification: \( (m - 4)/(4 - m) = -1 \).
• Finally, the simplified expression becomes \( -(m + 5) \) or \( -m - 5 \).

Simplification helps to express the result in the most streamlined manner possible, making the solution clearer and easier to interpret. It plays a significant role in understanding complex problems by focusing on core components of an expression or equation.