Problem 14 - Beginning and Intermediate Algebra

Question

Divide. $$\left(60 z^{5}+3 z^{4}-10 z\right) \div\left(5 z^{2}\right)$$

Step-by-Step Solution

Answer

The final result of the division is \(12z^3 + \frac{3}{5}z^2 - 2z^{-1}\).

Step 1: Identify the polynomial and monomial

The first step is to identify the polynomial and the monomial. In our case, the polynomial is \(\left(60z^5 + 3z^4 - 10z\right)\) and the monomial is \(\left(5z^2\right)\).

Step 2: Divide each term of the polynomial by the monomial

Next, we will divide each term of the polynomial by the given monomial. We will do the following operations: \[ \begin{aligned} \frac{60z^5}{5z^2} \\ \frac{3z^4}{5z^2} \\ \frac{-10z}{5z^2} \end{aligned} \]

Step 3: Simplify the resulting terms

After dividing, we will simplify each resulting term by using the rules of exponents as well as arithmetic operations. We have: \[ \begin{aligned} \frac{60z^5}{5z^2} &= \frac{60}{5} \cdot \frac{z^5}{z^2} = 12z^{3} \\ \frac{3z^4}{5z^2} &= \frac{3}{5} \cdot \frac{z^4}{z^2} = \frac{3}{5}z^{2} \\ \frac{-10z}{5z^2} &= \frac{-10}{5} \cdot \frac{z}{z^2} = -2z^{-1} \end{aligned} \]

Step 4: Write the simplified expression

Now that we have simplified the resulting terms, we can write the final expression by combining these terms. The final result of the division is: \[ 12z^3 + \frac{3}{5}z^2 - 2z^{-1} \]

Key Concepts

Monomial

A monomial is a single term algebraic expression, consisting of a coefficient and variables raised to a power. In simpler terms, it is an expression like \(5z^2\) from the original exercise, where "5" is the coefficient and \(z^2\) represents the variable "z" raised to the power of 2.

• Monomials can have more than one variable.
• The degree of a monomial is the sum of the exponents of all the variables in it.

For example, \(7xy^2\) is a monomial with a degree of 3, as the exponents for "x" and "y" total 3 (i.e., 1+2).
When performing polynomial division, understanding the concept of a monomial is important for dividing each term of the polynomial accordingly.

Simplifying Expressions

Simplifying expressions is a process of rewriting them in a more manageable form. It helps in reducing the complexity of algebraic expressions and making calculations easier.
To simplify, follow these steps:

• Divide the coefficients (numerical parts of each term).
• Apply the rules of exponents on the variables.

Let's consider simplifying the term \(\frac{60z^5}{5z^2}\). First, divide 60 by 5, resulting in 12. Then, apply the rules of exponents to \(z^5\) divided by \(z^2\). This gives \(z^{5-2} = z^3\). Therefore, the term simplifies to \(12z^{3}\).
Simplification is vital to solving polynomial division, providing a more concise form of the expression.

Rules of Exponents

The rules of exponents are guidelines that simplify the calculation of powers. They are essential when dealing with terms in polynomial division, like in our exercise.
Here are some basic rules:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Zero Exponent: \(a^0 = 1\) (where \(aeq 0\))
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)

In the division \(\frac{z^5}{z^2}\), apply the quotient of powers rule to get \(z^{5-2} = z^3\). Another example from our solution is dealing with \(z^{-1}\) from \(\frac{-10z}{5z^2}\), where the negative exponent indicates that \(z^1\) can be expressed as \(\frac{1}{z}\).
Understanding these rules leads to the accurate simplification of expressions, simplifying terms correctly even when they involve complex expressions.