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Chapter 5: Problem 10

Perform each division. $$ \frac{12 t^{5}-6 t^{3}+6 t^{2}}{6 t^{2}} $$

Short Answer

Expert verified
\(2 t^{3} - t + 1\)

Step by step solution

01

Identify the common factor in the numerator and denominator

The numerator is \(12 t^{5}-6 t^{3}+6 t^{2}\) and the denominator is \(6 t^{2}\). Notice that each term in the numerator has a common factor of \(6 t^{2}\).
02

Rewrite the numerator in factored form

Rewrite the numerator \(12 t^{5}-6 t^{3}+6 t^{2}\) by factoring out \(6 t^{2}\): \(6 t^{2} (2 t^{3} - t + 1)\).
03

Simplify the fraction

Divide the factored numerator by the denominator: \[\frac{6 t^{2} (2 t^{3} - t + 1)}{6 t^{2}}\].
04

Cancel out the common factor

The \(6 t^{2}\) in the numerator and the denominator cancel each other out, resulting in: \[2 t^{3} - t + 1\].

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Key Concepts

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

Factoring Polynomials
Factoring helps in simplifying problems.
By reducing complex terms into simpler parts, it becomes easier to perform operations like division.
Simplifying Algebraic Expressions
  • Factor out the greatest common term.
  • Rewrite the expression.
  • Cancel out common terms.

This makes solving the problem much easier!
Common Factor
A common factor is a term that appears in every part of an expression.
In our exercise, both the numerator and the denominator share the common factor \(6 t^{2}\).
Identifying common factors is essential in simplifying algebraic expressions.
It reduces the problem to its simplest form.
Here's how it works:
  • Look at each term in the expression.
  • Identify the common variables and coefficients.
  • Factor them out.

It's like finding common ingredients in different recipes!

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Most popular questions from this chapter

Evaluate each expression for \(x=3\) $$ 2 x^{2}-3 x+10 $$

Use scientific notation to calculate the answer to each problem. In September of \(2009,\) the population of the United States was about 307.5 million. To the nearest dollar, calculate how much each person in the United States would have had to contribute in order to make one lucky person a trillionaire (that is, to give that person \(\$ 1,000,000,000,000) .\)

Subtract. \(5 t^{2}+2 t-6\) \(5 t^{2}-3 t-9\) \(\hline\)

Perform each division using the "long division" process. $$ \frac{5 z^{3}-z^{2}+10 z+2}{z+2} $$

Find each product. \((x+y)^{2}(x-y)^{2}\)
See all solutions

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