Problem 65 - Algebra for College Students

Question

Solve each equation. $$\frac{9}{x^{3}-1}-\frac{1}{x-1}=\frac{2}{x^{2}+x+1}$$

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -5\).

Step 2 - Simplify the Terms

Rewrite the equation using the factored form: \(\frac{9}{(x-1)(x^2 + x + 1)} - \frac{1}{x-1} = \frac{2}{x^2 + x + 1}\).

Step 3 - Combine the Fractions

Find a common denominator for all terms: \(\frac{9}{(x-1)(x^2 + x + 1)} - \frac{(x^2 + x + 1)}{(x-1)(x^2 + x + 1)} = \frac{2(x-1)}{(x-1)(x^2 + x + 1)}\).

Step 4 - Combine into a Single Fraction

Combine the fractions into a single fraction: \(\frac{9 - (x^2 + x + 1)}{(x-1)(x^2 + x + 1)} = \frac{2(x-1)}{(x-1)(x^2 + x + 1)}\).

Step 5 - Simplify the Numerator

Simplify the numerator on the left side: \(\frac{9 - x^2 - x - 1}{(x-1)(x^2 + x + 1)} = \frac{8 - x^2 - x}{(x-1)(x^2 + x + 1)}\).

Step 6 - Equate Numerators

Equate the numerators since the denominators are already equal: \(8 - x^2 - x = 2(x - 1)\).

Step 7 - Solve for x

Simplify and solve for \(x\): \(8 - x^2 - x = 2x - 2\). Combine like terms to get: \ -x^2 - 3x + 10 = 0\. We can solve this quadratic equation to find \(x\).

Step 8 - Solve the Quadratic Equation

Solve the quadratic equation \( -x^2 - 3x + 10 = 0\). Factor the equation: \( -(x^2 + 3x - 10) = 0 \: (x+5)(x-2)=0 \Longrightarrow x = -5, x=2\).

Step 9 - Verify the Solutions

Substitute \(x = -5\) and \(x = 2\) back into the original equation to verify if they satisfy it.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill when solving rational equations, since it allows us to simplify complicated expressions. In this exercise, the denominator \(x^3 - 1\) was factored into \( (x-1)(x^2 + x + 1) \). Recognizing patterns in polynomials helps in this process.

Use the following rules of thumb for common factoring cases:

• Difference of squares: \( a^2 - b^2 = (a-b)(a+b) \).
• Sum/Difference of cubes: \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \).

For more complex polynomials, look for common factors or use methods such as grouping.

Common Denominator

Finding a common denominator is essential when combining rational expressions. This ensures the fractions have a unified base, making it easier to add or subtract them.

In our problem, the denominators \(x-1 \) and \(x^2 + x +1\) needed to be unified. This was achieved by multiplying \( (x-1)(x^2 + x + 1) \) as a common denominator.

Here’s the trick:

• Identify the least common multiple (LCM) of the denominators.
• Rewrite each fraction with the LCM as the new denominator.

This allows you to perform arithmetic operations directly on the numerators.

Simplifying Fractions

Simplifying fractions involves reducing the numerator and the denominator to their smallest form. This is crucial when dealing with complex rational equations.

In our equation, the simplified form of the left side was: \(\frac{8 - x^2 - x}{(x-1)(x^2 + x + 1)} \).

Follow these steps to simplify:

• Combine like terms in numerators and denominators.
• Factor where possible to cancel out common terms.

Always check if further simplification is possible by canceling any common factors.

Quadratic Equations

Solving quadratic equations can often pop up in rational expressions, just like in our given problem. After simplifying, we ended up solving: \( -x^2 - 3x + 10 = 0 \).

The process involves:

• Rewriting the equation in standard form \( ax^2 + bx + c = 0 \).
• Factoring into two binomials, if possible.
• Using the quadratic formula when necessary: \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \)

For our specific instance, we factored \ (-x^2 - 3x + 10) \ to \ (x+5)(x-2)=0 \ and found the solutions: \ x = -5 \ and \ x = 2 \.