Problem 90 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.. $$Divide:$\frac{9 x^{2}-y^{2}+15 x-5 y}{3 x^{2}+x y+5 x} \div \frac{3 x+y}{9 x^{3}+6 x^{2} y+x y^{2}}$$

Step-by-Step Solution

Answer

The answer is the simplified form of the fraction obtained in Step 3.

Step 1: Rewrite the division as multiplication

Rewrite the division of fractions as multiplication using the formula \(a/b ÷ c/d = a/b * d/c\). Consequently, the expression becomes \(\frac{9x^{2}-y^{2}+15x-5y}{3x^{2}+xy+5x} * \frac{9x^{3}+6x^{2}y+xy^{2}}{3x+y}\).

Step 2: Simplify the multiplication

Next, perform the multiplication of the two fractions. This is done by multiplying the numerators together and the denominators together. The result is \(\frac{(9x^{2}-y^{2}+15x-5y) * (9x^{3}+6x^{2}y+xy^{2})}{(3x^{2}+xy+5x) * (3x+y)}\).

Step 3: Factorize and simplify

Factorize the numerator and the denominator. Check if there are any common factors in the numerator and the denominator that can be cancelled out. If there are any, cancel them out to simplify the fraction.

Key Concepts

Fraction Multiplication and Division

When it comes to working with algebraic fractions, understanding multiplication and division is crucial. Multiplication of fractions is straightforward; you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. To divide fractions, however, you need to remember the key rule: dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is created by swapping the numerator and the denominator. For example, dividing by \(\frac{3}{4}\) is the same as multiplying by its reciprocal, \(\frac{4}{3}\).

In the exercise provided, dividing one algebraic fraction by another involves changing the division into multiplication by the reciprocal of the second fraction. This is the key step that transforms a division problem into a multiplication one, making it easier to handle the rest of the operations that follow, such as combining like terms and simplifying.

Factoring Algebraic Expressions

Factoring is the process of breaking down an expression into simpler terms or factors that, when multiplied together, give back the original expression. It's like taking a complex puzzle and separating it into its individual pieces, which are easier to analyze and work with. By factoring algebraic expressions, you can often find common factors in the numerator and denominator of a fraction. When you have these common factors, you can cancel them out to simplify the expression further.

For instance, in algebraic fractions, we might look for common binomials or recognize special products such as a difference of squares. It's important to become comfortable with different factoring techniques, including pulling out a common factor, using the FOIL method for binomials, and recognizing patterns such as the difference or sum of cubes. In our exercise, recognizing these factors can be the difference between a lengthy, complicated expression and a neat, simplified result.

Simplifying Algebraic Fractions

The final step after factoring expressions within algebraic fractions is to simplify. This means reducing the fractions to their simplest form by eliminating any common factors in the numerator and the denominator. It is equivalent to canceling out common factors in a numeric fraction like \(\frac{6}{8}\), which simplifies to \(\frac{3}{4}\) by dividing both the numerator and the denominator by 2.

In our exercise, after factoring, we look for any terms that appear in both the numerator and the denominator. If any are found, they can be canceled out, as anything divided by itself is 1. Remember that simplifying not only makes the expression easier to read but also easier to use in further calculations or applications. Be meticulous when simplifying, as overlooking a common factor can lead to a wrong answer or a less simplified form.