Problem 26 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 2. $$ \frac{m^{2}+7 m^{2}-7 m-49}{m^{3}} \cdot \frac{1}{m^{2}+7 m} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{(4m+7)(2m-7)}{m^{4}(m+7)} \).

Step 1: Simplify the Expression

Start by simplifying the expression \( \frac{m^{2}+7m^{2}-7m-49}{m^{3}} \). Combine like terms in the numerator: \( m^{2} + 7m^{2} = 8m^{2} \). This simplifies the expression to \( \frac{8m^{2} - 7m - 49}{m^{3}} \).

Step 2: Factor the Numerator

Factor the simplified numerator \( 8m^{2} - 7m - 49 \). To factor, seek two numbers whose product is \( 8 \times -49 = -392 \) and whose sum is \(-7\). Those numbers are \(-28\) and \(21\). Rewrite the expression as \( 8m^{2} - 28m + 21m - 49 \) and factor by grouping: \( 4m(2m - 7) + 7(2m - 7) \), leading to \((4m + 7)(2m - 7)\).

Step 3: Substitute in the Full Expression

Replace the factored numerator in the initial expression: \( \frac{(4m+7)(2m-7)}{m^{3}} \cdot \frac{1}{m^{2}+7m} \).

Step 4: Simplify the Denominator

Factor the second denominator \( m^{2} + 7m \) which becomes \( m(m+7) \). Update the expression: \( \frac{(4m+7)(2m-7)}{m^{3}} \cdot \frac{1}{m(m+7)} \).

Step 5: Combine and Simplify Fractions

Combine the two fractions: \( \frac{(4m+7)(2m-7)}{m^{3}m(m+7)} \). Simplify the fraction by canceling out terms that appear in both the numerator and denominator. There are no common terms to cancel directly here.

Step 6: Write the Final Expression

The final expression, fully simplified, is \( \frac{(4m+7)(2m-7)}{m^{4}(m+7)} \).

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that involves breaking down a polynomial into a product of simpler polynomials. This process is akin to finding the "building blocks" of a polynomial, which makes it easier to solve equations and simplify expressions.

To factor a polynomial, you generally look for pairs of numbers that multiply to give the product of the leading coefficient and the constant term while adding up to give the middle coefficient. For instance, in the polynomial \(8m^2 - 7m - 49\), we need numbers that multiply to \(-392\) (i.e., \(8 \times -49\)) and sum to \(-7\).

Through this process, we get \(-28\) and \(21\), which allows us to rewrite the equation as \(8m^2 - 28m + 21m - 49\). We then factor by grouping, pulling out common factors from pairs of terms to get \((4m + 7)(2m - 7)\). This illustrates two key strategies in factoring:

• Finding numbers that multiply and add to specific values.
• Using grouping to simplify and clearly factorize the expression.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator is a polynomial. Simplifying these expressions involves operations similar to those on numerical fractions, like finding common denominators and canceling terms.

In the given problem, we start with a complex expression \(\frac{8m^2 - 7m - 49}{m^3} \cdot \frac{1}{m^2 + 7m}\). Simplifying rational expressions involves reducing both the numerator and the denominator to their simplest forms by factoring and canceling common terms.

After factoring the numerator to \((4m + 7)(2m - 7)\) and the second denominator \(m(m + 7)\), the expression becomes smoother to handle. This simplicity is crucial for multiplying or dividing rational expressions efficiently.

Thus, simplifying rational expressions revolves around:

• Factoring polynomials to reveal common factors.
• Reducing both numerators and denominators to their simplest forms.

Understanding these fundamentals paves the way for seamlessly working with polynomial fractions.

Combining Like Terms

Combining like terms is a fundamental process in simplifying algebraic expressions. It involves collecting terms in a polynomial equation that have the same variables raised to the same powers.

In the original exercise, this concept is applied to the expression \(m^2 + 7m^2 - 7m - 49\). Here, the like terms \(m^2\) and \(7m^2\) combine to form \(8m^2\).

This operation simplifies the equation significantly, reducing complex expressions into more manageable ones. The goal when combining like terms is to streamline the equation by:

• Identifying terms with identical variable components.
• Adding or subtracting the coefficients of these terms.

For example, in the numerator of the initial expression, combining like terms transforms it into \(8m^2 - 7m - 49\), a much cleaner version that is easier to factor.