Problem 8 - Algebra for College Students

Question

Perform the indicated divisions of polynomials by monomials. $$ \frac{14 x y-16 x^{2} y^{2}-20 x^{3} y^{4}}{-x y} $$

Step-by-Step Solution

Answer

The quotient is \(-14 + 16xy + 20x^2y^3\).

Step 1: Distribute the Division

First, distribute the division across each term in the numerator. Each term in the polynomial \(14xy - 16x^2y^2 - 20x^3y^4\) will be divided by the monomial \(-xy\). This will give us three separate fractions: \[ \frac{14xy}{-xy}, \quad \frac{-16x^2y^2}{-xy}, \quad \frac{-20x^3y^4}{-xy} \]

Step 2: Simplify Each Fraction

Now, simplify each fraction by cancelling common factors:- The first term: \(\frac{14xy}{-xy} = -14\), since the \(xy\) cancels out in both numerator and denominator.- The second term: \(\frac{-16x^2y^2}{-xy} = 16x^{2-1}y^{2-1} = 16xy\) since one \(x\) and one \(y\) cancel.- The third term: \(\frac{-20x^3y^4}{-xy} = 20x^{3-1}y^{4-1} = 20x^2y^3\) since one \(x\) and one \(y\) cancel.

Step 3: Write the Final Expression

Assembling the simplified terms will give us the final answer. The terms are now:\[-14 + 16xy + 20x^2y^3\]

Key Concepts

Understanding Monomials

A monomial, in algebra, is a type of polynomial with only one term. It can include numbers, variables, or the product of numbers and variables. For example, in our exercise, the expression

• \(-xy\)

is a monomial because it consists of a single term made from the multiplication of the variables \(x\) and \(y\), with an implicit multiplicative coefficient of -1.
Monomials are common in mathematics due to their simplicity and their role in larger polynomial structures. When working with monomials, it's vital to be attentive to their coefficients and the exponents of each variable since they determine how interactions with other polynomials will proceed.
A crucial aspect of dealing with monomials and polynomials involves operations such as addition, subtraction, multiplication, and division. While addition and subtraction require like terms, division, as shown in our exercise, involves applying the division to each term separately.

Simplifying Fractions Involving Polynomials and Monomials

Simplifying fractions is a crucial skill when working with polynomials divided by monomials. Each term in the polynomial fraction is addressed individually. To simplify, first look for common factors in terms within both the numerator and the denominator.
Let's take an example from the given problem. The fraction

• \(\frac{14xy}{-xy}\)

is simplified by identifying that \(-xy\) is a common factor in both the numerator and the denominator. Canceling the common factor gives us

• \(-14\).

Performing similar operations on each of the polynomial's terms helps streamline the expression by reducing it to its simplest form. Remember to apply the rules of exponents carefully: when variables in the denominator match those in the numerator, subtract their exponents during the simplification.

Canceling Common Factors

Canceling common factors is essential to simplifying algebraic expressions. The process involves reducing expressions by eliminating factors that appear in both the numerator and the denominator. For polynomial division, this means looking at each term separately to find and cancel common factors.
For instance, consider the fraction

• \(\frac{-20x^3y^4}{-xy}\)

You'll see that both the \(-xy\) in the denominator and the \(-20x^3y^4\) in the numerator share variables \(x\) and \(y\). To cancel these, you subtract the exponents of the common variables:

• the exponent of \(x\) becomes \(3 - 1 = 2\)
• and the exponent of \(y\) becomes \(4 - 1 = 3\).

This leaves

• \(20x^2y^3\)

as the simplified term. Utilizing this strategy of canceling common factors across individual terms can drastically simplify and clarify polynomial expressions.