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Chapter 4: Problem 5

Divide. $$ \frac{15 x^{3}-10 x^{2}+5}{5} $$

Short Answer

Expert verified
The simplified result is \( 3x^{3} - 2x^{2} + 1 \).

Step by step solution

01

Simplify the fraction

Start by simplifying the fraction given in the problem: \( \frac{15 x^{3}-10 x^{2}+5}{5} \). Divide each term in the numerator by the denominator.
02

Divide each term

Divide each term in the numerator by the denominator one by one: \( \frac{15 x^{3}}{5} \), \( \frac{10 x^{2}}{5} \), and \( \frac{5}{5} \).
03

Simplify each division

\( \frac{15 x^{3}}{5} = 3 x^{3} \), \( \frac{10 x^{2}}{5} = 2 x^{2} \), and \( \frac{5}{5} = 1 \).
04

Write the final simplified form

Combine the simplified terms to get the final answer: \( 3 x^{3} - 2 x^{2} + 1 \).

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Key Concepts

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

Simplifying Fractions
Simplifying fractions means reducing a fraction to its simplest form.
This process makes the fraction easier to understand and work with.
For example, simplifying a fraction like \( \frac{15x^{3} - 10x^{2} + 5}{5} \) helps us handle it more effectively.
It involves dividing each term in the numerator by the denominator.
Doing this for each term in the example, we get:
  • \(\frac{15 x^{3}}{5} = 3 x^{3}\)
  • \(\frac{10 x^{2}}{5} = 2 x^{2}\)
  • \(\frac{5}{5} = 1\)

This makes the expression much simpler: \(3 x^{3} - 2 x^{2} + 1\).
Simplifying fractions is an essential skill in algebra, ensuring that we can analyze and solve problems more reliably.
Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operations.
Take the expression \(15x^{3} - 10x^{2} + 5\) as an example.
Here, the terms include a constant number and variables raised to powers.
An algebraic expression might have:
  • Constants (e.g., 5)
  • Variables (e.g., \(x\))
  • Co-efficients (e.g., 15 in \(15x^{3}\))
  • Operations (e.g., addition or subtraction)

Understanding algebraic expressions is crucial for simplifying, factoring, and solving equations.
A good grasp of these allows you to manipulate and simplify mathematical problems seamlessly.
Numerator and Denominator
A fraction consists of two parts: the numerator and the denominator.
The numerator is the top part, representing the number of parts considered.
The denominator is the bottom part, showing the total number of equal parts.
For instance, in the fraction \(\frac{15x^{3} - 10x^{2} + 5}{5}\),
  • The numerator is \(15x^{3} - 10x^{2} + 5\)
  • The denominator is 5.

Dividing both parts helps simplify the fraction.
By dividing each term in the numerator by the denominator, you effectively make the fraction easier to work with.

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Most popular questions from this chapter

Write using only positive exponents and then evaluate. Assume that all variables represent nonzero real numbers. $$ \left(\frac{1}{5 x}\right)^{-3} $$

Find each product. $$ (a-3)^{2} $$

Find each product. $$ (x+1)^{2} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(4^{-4}\right)^{-2} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(k^{2}\right)^{-3} k^{4} $$
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