Problem 26 - Algebra for College Students

Question

Solve each equation. $$ \frac{2}{n^{2}+4 n}+\frac{3}{n^{2}-3 n-28}=\frac{5}{n^{2}-6 n-7} $$

Step-by-Step Solution

Answer

The solution to the equation is \(n = 0\).

Step 1: Identify and Factor the Denominators

First, look at each denominator to simplify the equation. The denominators are \(n^2 + 4n\), \(n^2 - 3n - 28\), and \(n^2 - 6n - 7\). Factor each expression:- \(n^2 + 4n = n(n + 4)\).- \(n^2 - 3n - 28 = (n - 7)(n + 4)\).- \(n^2 - 6n - 7 = (n - 7)(n + 1)\).

Step 2: Find the Least Common Denominator (LCD)

The least common denominator is derived from the factored forms, taking the highest power of each factor present:\[ \text{LCD} = n(n + 4)(n - 7)(n + 1) \]

Step 3: Multiply Each Term by the LCD

Multiply every term in the equation by \(n(n + 4)(n - 7)(n + 1)\) to eliminate the denominators. Each term becomes:- \(\frac{2}{n(n + 4)} \, \rightarrow \, 2(n - 7)(n + 1)\)- \(\frac{3}{(n - 7)(n + 4)} \, \rightarrow \, 3n(n + 1)\)- \(\frac{5}{(n - 7)(n + 1)} \, \rightarrow \, 5n(n + 4)\)

Step 4: Simplify the Equation

The equation simplifies to:\[ 2(n - 7)(n + 1) + 3n(n + 1) = 5n(n + 4) \]Expand each term:- \(2(n - 7)(n + 1) = 2(n^2 - 6n - 7)\). - \(3n(n + 1) = 3n^2 + 3n\).- \(5n(n + 4) = 5n^2 + 20n\).

Step 5: Combine and Rearrange Terms

Combine all the terms on the left:\[ 2n^2 - 14n + 3n^2 + 3n = 5n^2 + 20n \]This simplifies to:\[ 5n^2 - 11n = 5n^2 + 20n \]

Step 6: Solve for Variable

Subtract \(5n^2\) from both sides to eliminate \(n^2\):\[ -11n = 20n \]Add 11n to both sides:\[ 0 = 31n \]Divide both sides by 31 to solve for \(n\):\[ n = 0 \]

Step 7: Verify the Solution

Substitute \(n = 0\) back into the original equation to verify. It should satisfy the equation without leading to undefined expressions in the denominators. The original equation results in valid expressions when \(n = 0\) is substituted, confirming the solution.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental technique in algebra. It involves breaking down polynomial expressions into products of simpler polynomials. This is highly useful when simplifying equations, especially for identifying common denominators.

For instance, in the given equation, the denominators are polynomials: \(n^2 + 4n\), \(n^2 - 3n - 28\), and \(n^2 - 6n - 7\). Factoring these expressions requires finding patterns or using methods like grouping or the quadratic formula.

Here's how you can factor them:

• For \(n^2 + 4n\): Factor by pulling out the common factor \(n\) to get \(n(n + 4)\).
• For \(n^2 - 3n - 28\): Recognize it as a trinomial and factor it into \((n - 7)(n + 4)\).
• For \(n^2 - 6n - 7\): Similarly, factor this trinomial into \((n - 7)(n + 1)\).

Once these polynomials are factored, it becomes easier to handle them during operations such as finding a least common denominator or solving equations.

Least Common Denominator

The least common denominator (LCD) is essential when dealing with rational expressions. It's used to combine fractions by converting them into like terms.

In the context of the exercise, the denominators were factored to be \(n(n + 4)\), \((n - 7)(n + 4)\), and \((n - 7)(n + 1)\). To find the LCD, you include each factor the highest number of times it appears in any one factorization.

This process can be outlined as:

• Find the distinct factors: \(n\), \(n + 4\), \(n - 7\), and \(n + 1\).
• Take the highest power of each factor found: All these factors appear only once in their respective terms.

Therefore, the least common denominator is \(n(n + 4)(n - 7)(n + 1)\). Using the LCD allows you to combine the fractions into a single expression. This makes further manipulation simpler and more straightforward.

Solving Rational Equations

Rational equations are equations that include fractions whose numerators and/or denominators are polynomials. Solving these involves eliminating the fractions by multiplying all terms by the least common denominator (LCD).

In our example, multiplying each term by \(n(n + 4)(n - 7)(n + 1)\) clears the denominators and results in a polynomial equation.

Here's what happens step-by-step:

• Identify the LCD: Use it to multiply each term in the equation.
• Clear fractions: The equation transforms, simplifying to traditional polynomial form.
• Solve: Handle it like any other polynomial equation, using algebraic methods to find the value of the variable.

This approach allows for the original rational equation to be tackled efficiently, bypassing the complications of dealing directly with fractions.

Polynomial Expressions

A polynomial expression consists of variables raised to natural number exponents and their coefficients. These forms the basics of algebraic manipulation and appear frequently in engineering, physics, and more.

In equations, polynomial expressions can vary in complexity. The original exercise showcases polynomials like \(n^2 + 4n\) and \(n^2 - 6n - 7\). Working with polynomial expressions requires operations such as:

• Adding or subtracting to combine like terms.
• Multiplying to expand factors into expanded terms.
• Factoring to simplify or solve equations.

Understanding the structure and behavior of polynomial expressions is crucial. It allows you to transform complex equations into simpler forms where solutions become evident. Mastering polynomial operations empowers algebraic problem-solving skills.