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Chapter 6: Problem 443

Divide each polynomial by the monomial. $$\frac{30 b+75}{5}$$

Short Answer

Expert verified
The solution is \(6b + 15\).

Step by step solution

01

Identify the terms in the polynomial

The polynomial given is \(30b + 75\). This polynomial has two terms: \(30b\) and \(75\).
02

Identify the monomial

The monomial given is \(5\).
03

Divide each term of the polynomial by the monomial

Divide each term of the polynomial \(30b + 75\) by the monomial \(5\). This can be written as: \[\frac{30b}{5} + \frac{75}{5}\].
04

Simplify each division

Simplify \(\frac{30b}{5}\) which equals \(6b\) and \(\frac{75}{5}\) which equals \(15\).
05

Combine the simplified terms

Combine the simplified results to get: \(6b + 15\).

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Key Concepts

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

Terms in a Polynomial
A polynomial is an expression that consists of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication.
It is crucial to identify different 'terms' in a polynomial because these are individual building blocks that help in performing various operations like addition, subtraction, and division.
For example, in the polynomial given in the exercise, \(30b + 75\), there are two terms: \(30b\) and \(75\).
Each term consists of:
  • Numerical coefficient: It's the number multiplying the variable (e.g., 30 in \(30b\)).
  • Variable: The letter that represents an unknown value (e.g., 'b' in \(30b\)).
  • Constant: A term with no variable, like 75.
Identifying them correctly is the first step to solving polynomials.
Monomial Division
Monomial division involves dividing a polynomial by a single term known as a monomial.
This process might sound tricky, but it is straightforward when broken down into smaller steps.
Using our example, \(\frac{30b + 75}{5}\), the monomial here is 5.
Here's how you do it:
  • Divide each term of the polynomial by the monomial separately. For \(30b + 75\), you divide \(\frac{30b}{5}\) and \(\frac{75}{5}\).
  • Simplify each fraction separately. For \(30b \div 5 = 6b\) and \(75 \div 5 = 15\).
By following these simple steps, you get the solution easily as \(6b + 15\).
Simplification of Fractions
Simplifying fractions is crucial in mathematics as it makes expressions neater and easier to work with.
It merely means reducing the fraction to its simplest form. In the context of our example \(\frac{30b}{5}\), the main task is to find the greatest common divisor (GCD) of the numerator and denominator.
Here are the steps:
  • Identify the GCD of the numerator and denominator. For \(30b\) and 5, the GCD is 5.
  • Divide both the numerator and the denominator by the GCD. Here, \(\frac{30b}{5} = 6b\).
  • Continue for any other terms, like \(\frac{75}{5} = 15\).
By practicing these steps, you can simplify virtually any fraction effortlessly and the final outcome is clear and simplified, as noted in the exercise: \(6b + 15\).

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