Problem 69 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{4 x}{x^{2}+6 x+8}+\frac{3}{x^{2}+x-6}+\frac{x-1}{x^{2}+x-12} $$

Step-by-Step Solution

Answer

Now, create a short answer based on the solution provided above: To add the given rational expressions, first factor the denominators and find the least common denominator (LCD), which is $(x+4)(x+2)(x-2)(x+3)(x-3)$. Rewrite each fraction with the LCD as the denominator and combine the rational expressions by adding the numerators. Finally, simplify the numerator by expanding the terms and combining like terms, resulting in the fully simplified expression: $\frac{6x^{4} -7x^{3} +9x^{2} -3x -126}{(x+4)(x+2)(x-2)(x+3)(x-3)}$.

Step 1: Factor the denominators

Factoring the denominators of each rational expression will allow us to find a common denominator more easily. $$ \frac{4x}{(x+4)(x+2)}+\frac{3}{(x-2)(x+3)}+\frac{x-1}{(x-3)(x+4)} $$

Step 2: Find the least common denominator (LCD)

Identify the factors in each denominator and look for the least common multiple of each factor. In this case, the LCD is: $$ (x+4)(x+2)(x-2)(x+3)(x-3) $$

Step 3: Rewrite each fraction with the LCD as the denominator

First, we need to figure out how to adjust each fraction to have the common denominator (LCD). Once we know that, we can rewrite each fraction with the LCD as the denominator. $$ \frac{4x}{(x+4)(x+2)}\times\frac{(x-2)(x+3)(x-3)}{(x-2)(x+3)(x-3)}\\ +\frac{3}{(x-2)(x+3)}\times\frac{(x+4)(x+2)(x-3)}{(x+4)(x+2)(x-3)}\\ +\frac{x-1}{(x-3)(x+4)}\times\frac{(x+2)(x-2)(x+3)}{(x+2)(x-2)(x+3)} $$

Step 4: Combine the rational expressions

Now that all fractions have the common denominator, we can add the numerators together to combine the fractions. $$ \frac{4x(x-2)(x+3)(x-3)+3(x+4)(x+2)(x-3)+(x-1)(x+2)(x-2)(x+3)}{(x+4)(x+2)(x-2)(x+3)(x-3)} $$

Step 5: Simplify the numerator

To simplify the numerator, we are going to expand all the terms and then combine like terms. $$ \frac{ 4x(x^{2} -x -6)(x^{2} -9) +3(x^{2}+6x+8)(x^{2} -9) +(x-1)(x^{2} -4x +4)(x^{2} +x -6) }{(x+4)(x+2)(x-2)(x+3)(x-3)} $$ Expand the numerator but don't expand the denominator. $$ \frac{ 4x(x^{4}-9x^{2} -x^{3}+9x +6x^{2} -54) +3(x^{4}-9x^{2} +6x^{3}-54x +8x^{2} -72) +(x^{4} -3x^{3} +2x^{2} -x^{3} +3x^{2} -2x) }{(x+4)(x+2)(x-2)(x+3)(x-3)} $$ Combine like terms in the numerator. $$ \frac{ 6x^{4} -7x^{3} +9x^{2} -3x -126 }{(x+4)(x+2)(x-2)(x+3)(x-3)} $$ Now the expression is fully simplified: $$ \frac{6x^{4} -7x^{3} +9x^{2} -3x -126}{(x+4)(x+2)(x-2)(x+3)(x-3)} $$

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra, particularly when dealing with rational expressions. To factor a polynomial, you look for the greatest common factors of the terms, or you search for special patterns, such as the difference of squares, perfect square trinomials, or trinomials that factor into binomials.

Consider the denominator in the given exercise, such as \(x^2 + 6x + 8\). It factors into \((x+4)(x+2)\) because \(x+4\) and \(x+2\) are the binomials whose product gives the original quadratic polynomial. The mastery of this process simplifies the complexity of adding and subtracting rational expressions, as it allows us to deal with simpler, broken-down parts of an expression.

Least Common Denominator

Finding the least common denominator (LCD) is akin to finding common ground. When dealing with fractions, the LCD is the smallest expression that each of the denominators can divide into without leaving a remainder.

In the context of adding and subtracting rational expressions, identifying the LCD is vital as it allows us to combine these expressions. It involves first factoring the polynomials in the denominators, then taking the highest power of each unique factor across them to form the LCD. For example, from the denominators \((x+4)(x+2)\), \((x-2)(x+3)\), and \((x-3)(x+4)\), we identify the LCD to be \((x+4)(x+2)(x-2)(x+3)(x-3)\). This step sets the stage for rewriting the fractions over a common base, making addition or subtraction possible.

Simplifying Algebraic Expressions

The essence of simplifying algebraic expressions lies in making them easier to understand and work with. When expressions are simplified, like terms are combined and any arithmetic performed.

After finding the LCD and adjusting each fraction in our problem, the next step is to express them with the common denominator. This transformation often expands the numerators, which might seem more complex at first. Ultimately, simplification is achieved by expanding the products in the numerators and then combining like terms to reach the most reduced form of the expression. It’s this clean-up stage that often presents a challenge and requires careful distribution and consolidation of terms.

Combining Like Terms

Combining like terms is a process of addition or subtraction that you can perform only on terms that have exactly the same variable parts. For example, \(2x\) and \(3x\) are like terms because they both contain the variable \(x\) to the same power. Meanwhile, \(x^2\) and \(x\) are not like terms because their variables are not of the same power.

In the provided exercise, after expanding all the numerators, you're faced with a polynomial that has many terms. You should gather all the like terms—those with the same variables raised to the same powers—and combine them to simplify the expression. For instance, if you encounter \(4x^3\) and \(-5x^3\), you’d combine them to get \(-x^3\). This vital step helps us crunch down complex expressions into their simplest algebraic form.