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Chapter 5: Problem 34

Perform each division. $$ \left(36 y^{2}-12 y^{3}+20 y\right) \div\left(4 y^{2}\right) $$

Short Answer

Expert verified
9 - 3y + \( \frac{5}{y} \)

Step by step solution

01

Write down the expression

The expression to be divided is \( 36y^{2} - 12y^{3} + 20y \) by \( 4y^{2} \).
02

Divide each term separately

Divide each term in the numerator by the denominator separately. We have: 1. \( \frac{36y^{2}}{4y^{2}} \)2. \( \frac{-12y^{3}}{4y^{2}} \)3. \( \frac{20y}{4y^{2}} \)
03

Simplify each fraction

Simplify each fraction individually: 1. \( \frac{36y^{2}}{4y^{2}} = 9 \)2. \( \frac{-12y^{3}}{4y^{2}} = -3y \)3. \( \frac{20y}{4y^{2}} = \frac{5}{y} \)
04

Combine the simplified terms

Combine the simplified terms to get the final result: \( 9 - 3y + \frac{5}{y} \)

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Key Concepts

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

Algebra
In algebra, we often deal with equations and expressions involving variables. Here, we are focusing on polynomial division. A polynomial is an expression that consists of variables, coefficients, and exponents. For example, in the exercise we have the polynomial \(36y^{2} - 12y^{3} + 20y\).
When performing polynomial division, we divide each term of the polynomial by the given divisor separately.
This is similar to dividing numbers individually in arithmetic, but involves handling variables and their powers.
Simplifying Fractions
To simplify fractions, you break them down to their simplest form. In the context of polynomial division, it involves reducing each term by the common factors.
For example:
  • \( \frac{36y^{2}}{4y^{2}} \) can be reduced by dividing both the numerator and the denominator by \(4y^{2}\), resulting in \(9\).
  • \( \frac{-12y^{3}}{4y^{2}} \) becomes \(-3y\) after dividing both parts by \(4y^{2}\).
  • \( \frac{20y}{4y^{2}} \) becomes \( \frac{5}{y} \) after simplification.

Simplifying fractions in polynomials is a crucial step to make the final expression easier to work with.
Rational Expressions
A rational expression is a fraction in which both the numerator and the denominator are polynomials. Rational expressions can often be simplified by factoring and reducing them, much like regular numeric fractions.
In our exercise, after dividing and simplifying each term, we combine them to form the final rational expression:
  • \( \frac{36y^{2}}{4y^{2}} = 9 \)
  • \( \frac{-12y^{3}}{4y^{2}} = -3y \)
  • \( \frac{20y}{4y^{2}} = \frac{5}{y} \)
The final rational expression is \(9 - 3y + \frac{5}{y}\). Understanding how to work with rational expressions is important for solving more complex algebraic problems efficiently.

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Most popular questions from this chapter

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