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Chapter 7: Problem 47

Divide as indicated. $$\frac{y^{2}-y}{15} \div \frac{y-1}{5}$$

Short Answer

Expert verified
\(\frac{y}{3}\)

Step by step solution

01

Break down the expression

Rewrite the expression: \( \frac{y^{2}-y}{15} \div \frac{y-1}{5} \) as \( \frac{y^{2}-y}{15} \times \frac{5}{y-1} \). The division of fractions is the same as multiplication by the reciprocal of the second fraction.
02

Factor out common factors

You can factor \(y\) out of the numerator of the first fraction: \( \frac{y(y-1)}{15} \times \frac{5}{y-1} \). Now both fractions share a common factor of \(y - 1\).
03

Simplify the Expression

By cancelling out the common factors, you get: \( \frac{y}{15} \times 5\). This simplifies further to: \( \frac{y}{3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Division
When dividing fractions, the operation might seem tricky at first, but it follows a simple rule: multiply by the reciprocal.
In our example, we had to divide \(\frac{y^{2}-y}{15}\) by \(\frac{y-1}{5}\).
Instead of performing division directly, we take the reciprocal of the second fraction and change the division operation into multiplication.
  • Original operation: \(\frac{y^{2}-y}{15} \div \frac{y-1}{5}\)
  • Revamped as multiplication: \(\frac{y^{2}-y}{15} \times \frac{5}{y-1}\)
This approach applies to any fraction division problem, ensuring consistency and simplicity.
Understanding this fundamental step sets the stage for more complex operations like working with Rational Expressions.
Rational Expressions
Rational expressions are essentially fractions that include polynomials in the numerator and the denominator.
Just like numerical fractions, we can perform operations on them such as addition, subtraction, multiplication, and division.
In this situation, we are dealing with the rational expression \(\frac{y^{2}-y}{15}\). This means our numerator \(y^{2}-y\) is a polynomial.
  • Be sure to simplify expressions when possible, factoring plays a key role in this step.
  • Our goal often includes cancelling terms to reach a simpler form.
With \(\frac{y^{2}-y}{15} \times \frac{5}{y-1}\), we noticed a common term \((y-1)\) which allowed simplification.
Recognizing and eliminating factors common to both sides is crucial.
This process refines the expression without altering its value, making calculations simpler and more manageable.
Factoring Polynomials
Factoring polynomials is a crucial skill to master when dealing with algebraic expressions.
It involves breaking down a complex polynomial into simpler 鈥榝actorable鈥� parts.
In our example, the polynomial in the numerator \(y^{2}-y\) was factored into \(y(y-1)\).
  • You look for common factors initially to make expressions manageable.
  • This helped simplify later operations by revealing common terms for cancellation.
By factoring, calculations reduce in complexity, allowing the simplification of those daunting rational expressions.
It's always about simplifying the problem at its root, slowly chipping away at the expression layer by layer. This approach doesn't just apply to simplifying; it smoothens solving equations and aids graphing polynomials as well.

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Most popular questions from this chapter

In Exercises \(94-96,\) use a graphing utility to solve each rational equation. Graph each side of the equation in the given viewing rectangle. The first coordinate of each point of intersection is a solution. Check by direct substitution. $$\begin{aligned} &x+\frac{6}{x}=-5\\\ &[-10,10,1] \text { by }[-10,10,1] \end{aligned}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y^{2}-1}+\frac{2 y}{y-y^{2}}$$

Add or subtract as indicated. Simplify the result, if possible. $$7+\frac{1}{x-5}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{9 x+3}{x^{2}-x-6}+\frac{x}{3-x}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x+7}{4 x+12}+\frac{x}{9-x^{2}}$$
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