Subtracting Rational Expressions A Step-by-Step Guide

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In the realm of mathematics, particularly in algebra, the ability to manipulate and simplify rational expressions is a fundamental skill. Rational expressions, which are essentially fractions with polynomials in the numerator and denominator, frequently appear in various mathematical contexts. One common operation involving rational expressions is subtraction. This article delves into a step-by-step guide on how to subtract the given rational expressions:

$$\frac{7}{2w^2-19w+9} - \frac{4w+3}{2w^2+w-1}$$

We will explore the necessary techniques, including factoring, finding common denominators, and simplifying the resulting expressions. Mastering these techniques will not only enhance your algebraic proficiency but also provide a solid foundation for more advanced mathematical concepts.

Factoring the Denominators

The first crucial step in subtracting rational expressions involves factoring the denominators. Factoring allows us to identify common factors, which are essential for finding the least common denominator (LCD). Let's factor each denominator separately.

Factoring ${2w^2 - 19w + 9}$

To factor the quadratic expression ${2w^2 - 19w + 9}$, we look for two numbers that multiply to ${2 imes 9 = 18}$ and add up to -19. These numbers are -1 and -18. We can rewrite the middle term using these numbers:

$$2w^2 - 19w + 9 = 2w^2 - 18w - w + 9$$

Now, we factor by grouping:

$$2w^2 - 18w - w + 9 = 2w(w - 9) - 1(w - 9) = (2w - 1)(w - 9)$$

Thus, the factored form of ${2w^2 - 19w + 9}$ is ${(2w - 1)(w - 9)}$.

Factoring ${2w^2 + w - 1}$

Similarly, to factor the quadratic expression ${2w^2 + w - 1}$, we look for two numbers that multiply to ${2 imes -1 = -2}$ and add up to 1. These numbers are 2 and -1. We rewrite the middle term using these numbers:

$$2w^2 + w - 1 = 2w^2 + 2w - w - 1$$

Factoring by grouping gives us:

$$2w^2 + 2w - w - 1 = 2w(w + 1) - 1(w + 1) = (2w - 1)(w + 1)$$

Therefore, the factored form of ${2w^2 + w - 1}$ is ${(2w - 1)(w + 1)}$.

By successfully factoring the denominators, we transform the original expression into:

$$\frac{7}{(2w - 1)(w - 9)} - \frac{4w + 3}{(2w - 1)(w + 1)}$$

This factored form is crucial for the next step: finding the least common denominator.

Determining the Least Common Denominator (LCD)

Once the denominators are factored, the next step is to find the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. It is essential for combining the fractions into a single expression. To determine the LCD, we identify all unique factors present in the denominators and take the highest power of each factor.

In our case, the denominators are ${(2w - 1)(w - 9)}$ and ${(2w - 1)(w + 1)}$. The unique factors are ${(2w - 1)}$, ${(w - 9)}$, and ${(w + 1)}$. Since each factor appears only once in each denominator, the LCD is simply the product of these factors:

$$LCD = (2w - 1)(w - 9)(w + 1)$$

The least common denominator now allows us to rewrite each fraction with this common denominator, making it possible to perform the subtraction.

Rewriting Fractions with the LCD

To subtract rational expressions, it is necessary to rewrite each fraction with the least common denominator (LCD). This involves multiplying the numerator and the denominator of each fraction by the factors needed to obtain the LCD. Let's rewrite each fraction with the LCD we found in the previous step, which is ${(2w - 1)(w - 9)(w + 1)}$.

Rewriting the First Fraction

The first fraction is ${\frac{7}{(2w - 1)(w - 9)}}$. To get the LCD, we need to multiply the denominator by ${(w + 1)}$. Therefore, we also multiply the numerator by ${(w + 1)}$:

$$\frac{7}{(2w - 1)(w - 9)} imes \frac{(w + 1)}{(w + 1)} = \frac{7(w + 1)}{(2w - 1)(w - 9)(w + 1)}$$

This gives us the first fraction rewritten with the LCD.

Rewriting the Second Fraction

The second fraction is ${\frac{4w + 3}{(2w - 1)(w + 1)}}$. To get the LCD, we need to multiply the denominator by ${(w - 9)}$. Consequently, we multiply the numerator by ${(w - 9)}$ as well:

$$\frac{4w + 3}{(2w - 1)(w + 1)} imes \frac{(w - 9)}{(w - 9)} = \frac{(4w + 3)(w - 9)}{(2w - 1)(w + 1)(w - 9)}$$

Now, both fractions have the same denominator, allowing us to perform the subtraction.

Combining the Rewritten Fractions

After rewriting the fractions with the LCD, the original expression becomes:

$$\frac{7(w + 1)}{(2w - 1)(w - 9)(w + 1)} - \frac{(4w + 3)(w - 9)}{(2w - 1)(w - 9)(w + 1)}$$

Now that both fractions have the same denominator, we can combine them by subtracting the numerators.

Subtracting the Numerators

With both fractions now sharing a common denominator, the next step involves subtracting the numerators. This requires careful expansion and simplification of the expressions in the numerators. Let's proceed with the subtraction:

$$\frac{7(w + 1) - (4w + 3)(w - 9)}{(2w - 1)(w - 9)(w + 1)}$$

Expanding the Numerators

First, we expand the expressions in the numerator:

$$7(w + 1) = 7w + 7$$

$$(4w + 3)(w - 9) = 4w^2 - 36w + 3w - 27 = 4w^2 - 33w - 27$$

Now, substitute these expansions back into the numerator:

$$7w + 7 - (4w^2 - 33w - 27)$$

Distributing the Negative Sign

Next, distribute the negative sign across the terms in the second expression:

$$7w + 7 - 4w^2 + 33w + 27$$

Combining Like Terms

Now, combine like terms in the numerator:

$$-4w^2 + (7w + 33w) + (7 + 27) = -4w^2 + 40w + 34$$

Thus, the simplified numerator is ${-4w^2 + 40w + 34}$.

Rewriting the Expression

The expression now looks like this:

$$\frac{-4w^2 + 40w + 34}{(2w - 1)(w - 9)(w + 1)}$$

With the numerator simplified, we move on to the final step: simplifying the rational expression.

Simplifying the Rational Expression

After subtracting the numerators, we obtain the expression:

$$\frac{-4w^2 + 40w + 34}{(2w - 1)(w - 9)(w + 1)}$$

To simplify this rational expression, we first look for common factors in the numerator that can be factored out. Then, we check if any of these factors can cancel with factors in the denominator.

Factoring the Numerator

Let's factor the numerator ${-4w^2 + 40w + 34}$. We can start by factoring out a common factor of -2:

$$-2(2w^2 - 20w - 17)$$

The quadratic expression ${2w^2 - 20w - 17}$ does not factor easily using simple integer factors. We can check its discriminant to confirm this.

Checking the Discriminant

The discriminant ${\Delta}$ of a quadratic expression ${ax^2 + bx + c}$ is given by ${\Delta = b^2 - 4ac}$. For ${2w^2 - 20w - 17}$, we have:

$$\Delta = (-20)^2 - 4(2)(-17) = 400 + 136 = 536$$

Since 536 is not a perfect square, the quadratic expression does not have rational roots, and thus cannot be factored further using integers.

Analyzing the Denominator

The denominator is ${(2w - 1)(w - 9)(w + 1)}$. We need to check if any of these factors can cancel with the factored numerator, which is ${-2(2w^2 - 20w - 17)}$. Upon inspection, we see that there are no common factors between the numerator and the denominator.

Final Simplified Expression

Since we cannot simplify the rational expression further, the final simplified expression is:

$$\frac{-2(2w^2 - 20w - 17)}{(2w - 1)(w - 9)(w + 1)}$$

This is the simplest form of the result after subtracting the given rational expressions.

Conclusion

In this article, we have provided a comprehensive guide on how to subtract rational expressions, specifically addressing the problem:

$$\frac{7}{2w^2-19w+9} - \frac{4w+3}{2w^2+w-1}$$

We meticulously walked through each step, from factoring the denominators and finding the least common denominator (LCD) to rewriting the fractions with the LCD, subtracting the numerators, and simplifying the final expression. The key steps included:

1. Factoring the Denominators: Factoring ${2w^2 - 19w + 9}$ into ${(2w - 1)(w - 9)}$ and ${2w^2 + w - 1}$ into ${(2w - 1)(w + 1)}$.
2. Determining the LCD: Identifying the LCD as ${(2w - 1)(w - 9)(w + 1)}$.
3. Rewriting Fractions with the LCD: Multiplying the numerators and denominators to obtain equivalent fractions with the LCD.
4. Subtracting the Numerators: Expanding and simplifying the expression in the numerator.
5. Simplifying the Rational Expression: Factoring the numerator and checking for common factors with the denominator.

By following these steps, we arrived at the simplified expression:

$$\frac{-2(2w^2 - 20w - 17)}{(2w - 1)(w - 9)(w + 1)}$$

This detailed process not only provides the solution to the specific problem but also equips you with the skills to tackle similar problems involving rational expressions. Mastering these techniques is crucial for success in algebra and beyond. Remember, practice is key, so try applying these steps to various examples to solidify your understanding.