Problem 32 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3}{x-2}+\frac{4}{x+3}$$

Step-by-Step Solution

Answer

The solution to the given subtraction exercise is \(-\frac{3}{x}\).

Step 1: Identify the common denominator

Both fractions have the same denominator, \(2x^4\), which will as well serve as the common denominator.

Step 2: Subtract the numerators

Subtract the numerator of the second fraction from the numerator of the first fraction. This gives us \((x^3 - 3)-(7x^3 - 3)\).

Step 3: Simplify the numerator

Simplify the result from step 2 to get \(-6x^3\).

Step 4: Combine the result

Combine the simplified numerator and the common denominator to get the result, \(\frac{-6x^3}{2x^4}\).

Step 5: Simplify the result

Simplify the final expression to a lower term, and this gives \(-\frac{3}{x}\).

Key Concepts

Common Denominator

When working with fractions, whether they're numerical or algebraic, one key concept to understand is the common denominator. It's essentially the shared base that allows us to combine or compare fractions. Imagine trying to compare the sizes of different slices of pizza without referring to the size of the whole pizza - it would be confusing, right? A common denominator serves as that 'whole pizza' reference point.

In our exercise, the fractions \(\frac{x^{3}-3}{2x^{4}}\) and \(\frac{7x^{3}-3}{2x^{4}}\) both have the same denominator, \(2x^{4}\). This similarity is what makes subtraction possible without the need for any initial adjustments. Having a common denominator simplifies the process and ensures the terms remain in the same 'pizza size', or in mathematical terms, the same degree of x.

Numerators Subtraction

Moving from the concept of common denominators, we approach the next step - numerator subtraction. This is where the actual 'action' in subtracting polynomials happens. Think of it as taking pieces away from one 'pile' of algebraic expressions and seeing what you're left with.

In the equation we're given, the numerators \(x^3 - 3\) and \(7x^3 - 3\) are to be subtracted. To do this, we line up the terms so each term in one numerator corresponds to a similar term in the other. Terms with the same degree of x are grouped together to make the subtraction straightforward. The result is a single, simplified polynomial that represents the difference between the two numerators.

Simplifying Expressions

The heart of algebra lies in simplifying expressions. It's the process of turning a complicated expression into its simplest form. In our exercise, after subtracting the numerators, we are left with a new algebraic fraction. However, it's still not in its most simplified form.

Simplifying often involves combining like terms, factoring, and reducing fractions. For instance, once the numerators are subtracted, we need to combine the like terms to get a single expression for the numerator. This is then placed over the common denominator. If there are common factors in the numerator and the denominator, we reduce them to further simplify the expression. It's a bit like cleaning up your desk: you start with a bit of a mess and organize everything until it's neat and tidy.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both, contain algebraic expressions. They follow the same rules as regular fractions but require an understanding of polynomial operations. Simplifying algebraic fractions might involve reducing them to lower terms, factoring polynomials, and canceling out terms.

In our exercise, we end with the algebraic fraction \(\frac{-6x^3}{2x^4}\). By recognizing that \(x\) can be canceled out from the numerator and denominator, we simplify this to \(\frac{-3}{x}\), a much neater and more useful form for further mathematical operations or practical applications. It's important to note that final expressions in their simplest form provide greater clarity and often reveal more about the relationships between the algebraic quantities involved.