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Chapter 6: Problem 47

Write each rational expression in lowest terms. $$ \frac{3 m^{2}-3 m}{5 m-5} $$

Short Answer

Expert verified
\(\frac{3m}{5}\)

Step by step solution

01

Factor the numerator

Identify common factors in the numerator. The numerator is \(3m^2 - 3m\). The common factor in both terms is 3m, so factor out 3m: \[3m(m - 1)\]
02

Factor the denominator

Identify common factors in the denominator. The denominator is \(5m - 5\). The common factor in both terms is 5, so factor out 5: \[5(m - 1)\]
03

Simplify the expression

Now, the expression can be written with the factored terms: \[\frac{3m(m-1)}{5(m-1)}\]We can cancel out the common term \((m - 1)\) from the numerator and the denominator: \[\frac{3m}{5}\]

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into simpler 'factors' that, when multiplied together, give back the original polynomial.
For example in the given exercise, we started with the polynomial in the numerator, which is \(3m^2 - 3m\).
This polynomial can be factored by identifying a common factor for each term.
Here, we see that both terms share a common factor of \(3m\).
So, we can factor \(3m\) out of each term to get \(3m(m - 1)\).
Factoring helps us simplify polynomials, making it easier to work with them in algebraic expressions, especially in rational expressions.
Algebraic Fractions
Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions.
These are similar to numerical fractions but involve variables.
In our exercise, the original rational expression is \( \frac{3m^2 - 3m}{5m - 5} \).
The key to simplifying algebraic fractions is having a thorough understanding of factoring polynomials and canceling common factors.
Once we factor both the numerator and the denominator, we can often simplify the fraction by canceling out terms.
This is an essential skill in algebra that helps students work with complex expressions more efficiently.
Canceling Common Factors
Canceling common factors is a crucial step in simplifying rational expressions.
Once we have factored both the numerator and the denominator in our exercise, we had \( \frac{3m(m - 1)}{5(m - 1)} \).
We notice that \( (m - 1) \) is a common factor in both the numerator and the denominator.
By canceling out \( (m - 1) \) from both, we are left with \( \frac{3m}{5} \).
It's important to remember that we can only cancel factors, not terms.
This simplification makes the expression easier to interpret and work with.
One must always ensure that a common factor is present before attempting to cancel, as canceling terms improperly can lead to incorrect results.

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Most popular questions from this chapter

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{1+p^{3}}{1+p} $$

Multiply. Write each answer in lowest terms. $$ \frac{4(y-2)}{x} \cdot \frac{3 x}{6(y-2)^{2}} $$

Only one of these choices is equal to \(\frac{\frac{1}{3}+\frac{1}{12}}{\frac{1}{2}+\frac{1}{4}} .\) Which one is it? Answer this question without showing any work, and explain your reasoning. A. \(\frac{5}{9}\) B. \(-\frac{5}{9}\) C. \(-\frac{9}{5}\) D. \(-\frac{1}{12}\)

Multiply or divide. Write each answer in lowest terms. $$ \frac{r^{2}+r-6}{r^{2}+4 r-12} \div \frac{r+3}{r-1} $$

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{1}{2}+\frac{7}{8} $$
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