Problem 71 - Prealgebra and Introductory Algebra

Question

Divide. $$\frac{3 x^{2} y-9 x y}{a^{2} b} \div \frac{3 x^{2}-x^{3}}{a b^{2}}$$

Step-by-Step Solution

Answer

The answer is \(\frac{(3 x^{2} y-9 x y)*(a b^{2})}{(a^{2} b)*(3 x^{2}-x^{3})}\). The final result can vary depending on the simplification.

Step 1: Reciprocal of the Second Expression

Find the reciprocal of the second expression by swapping the numerator and denominator. The reciprocal of \(\frac{3 x^{2}-x^{3}}{a b^{2}}\) is \(\frac{a b^{2}}{3 x^{2}-x^{3}}\).

Step 2: Multiplication

Replace the division operation with multiplication of the first expression and the reciprocal of the second expression: \(\frac{3 x^{2} y-9 x y}{a^{2} b} \times \(\frac{a b^{2}}{3 x^{2}-x^{3}}\) \)

Step 3: Simplifying

Multiply the numerators together and multiply the denominators together: \(\frac{(3 x^{2} y-9 x y)*(a b^{2})}{(a^{2} b)*(3 x^{2}-x^{3})}\). In the end, you should simplify the expression step by step: factor out common terms if possible in the numerator and denominator, and then cancel out these common terms.

Key Concepts

Reciprocal of a Fraction

Understanding the concept of the reciprocal of a fraction is crucial when dividing polynomials. A reciprocal is simply a flipped version of the fraction, where the numerator becomes the denominator and vice versa. For any non-zero fraction, its reciprocal is found by exchanging its top and bottom parts.

For instance, if you have a fraction such as \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \), provided that neither \( a \) nor \( b \) is zero. In the context of dividing polynomials, we use this concept to transform a division problem into a multiplication problem, which is typically easier to solve.

In the given exercise, the reciprocal of \( \frac{3 x^{2}-x^{3}}{a b^{2}} \) is \( \frac{a b^{2}}{3 x^{2}-x^{3}} \) as shown in Step 1. Remember, the only restriction here is that none of the terms in the fraction can be zero, as divisions by zero are undefined.

Multiplying Polynomials

Multiplying polynomials is the next step that comes after finding the reciprocal of a fraction in the division process. This operation involves distributing each term of one polynomial over each term of the other polynomial.

To multiply polynomials, follow these quick steps:

• Start with one term of the first polynomial.
• Multiply it by each term of the second polynomial.
• Repeat this process for each term in the first polynomial.
• Finally, combine like terms to simplify the result.

In our exercise, once we replaced division with multiplication (in Step 2), we multiply the numerators together and the denominators together. This process is illustrated with the expression \( \frac{(3 x^{2} y-9 x y)*(a b^{2})}{(a^{2} b)*(3 x^{2}-x^{3})} \), which needs careful expansion and subsequent simplification.

Simplifying Algebraic Expressions

After multiplying the polynomials, the next goal is simplifying the algebraic expression. Simplifying is the process of making an algebraic expression as clean and concise as possible, often making it easier and more efficient to work with.

To simplify an algebraic expression:

• Look for common factors in the numerator and denominator.
• Factor these out.
• Cancel out common terms that appear in both the numerator and the denominator.
• Rewrite the result in its simplest form.

It’s important to recognize that terms can only be canceled if they appear as factors, not as addends. Simplification might require expanding polynomials first and then combining and reducing similar terms, as seen in the final steps of our exercise. The expression becomes simplest when no further factoring is possible and all common terms have been canceled.