Simplify the Rational Expression: (10x²-2x)/(5x-1)

Q: Why can't I just cancel the x terms directly?

You can only cancel complete factors, not individual terms! The expression $ 10x^2-2x $ has terms connected by subtraction, so you must factor first to reveal the common factor.

Q: How do I know when I can cancel something?

You can cancel when the same factor appears in both the numerator and denominator. After factoring, we got $ \frac{2x(5x-1)}{5x-1} $, and (5x-1) appears in both places.

Q: Do I need to worry about when x makes the denominator zero?

Yes! The original expression is undefined when $ 5x-1 = 0 $, so $ x \neq \frac{1}{5} $. Even after simplifying to 2x, this restriction still applies.

Q: How can I check if my factoring is correct?

Multiply out your factored form! If $ 2x(5x-1) $ gives you back $ 10x^2-2x $, then your factoring is correct.

Rational Expressions with Common Factor Reduction

Simplify the following expression:

$\frac{10x^2-2x}{5x-1}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 Let's break down 10 into factors 5 and 2

00:06 Let's break down the power into multiplications

00:15 Let's mark the common factors

00:36 Let's take out the common factors from the parentheses

00:47 Let's reduce what we can

00:54 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following expression:

$\frac{10x^2-2x}{5x-1}$

Step-by-step solution

Let's simplify the given expression:

$\frac{10x^2-2x}{5x-1}$ Remember that we can reduce complete expressions only when both the numerator and denominator are completely factored into multiplication expressions,

We must first use factorization, identify that in the numerator we can factor out a common term, then reduce the expressions in the resulting fraction:

$\frac{10x^2-2x}{5x-1} \\ \frac{2x(5x-1)}{5x-1} \\ \downarrow\\ \boxed{2x}$ Therefore, the correct answer is answer A.

Final Answer

$2x$

Key Points to Remember

Essential concepts to master this topic

Factorization Rule: Factor numerator completely before attempting to cancel terms
Technique: Factor out 2x from 10x²-2x to get 2x(5x-1)
Check: Multiply 2x by (5x-1) to verify: 2x(5x-1) = 10x²-2x ✓

Common Mistakes

Avoid these frequent errors

Canceling terms without complete factorization
Don't try to cancel x from 10x² and 5x directly = wrong answer like 2x-2! This ignores proper algebraic rules. Always factor the numerator completely first, then cancel only common factors that appear in both numerator and denominator.

Practice Quiz

Test your knowledge with interactive questions

Identify the field of application of the following fraction:

$\frac{7}{13+x}$

FAQ

Everything you need to know about this question

Why can't I just cancel the x terms directly?

You can only cancel complete factors, not individual terms! The expression $10x^2-2x$ has terms connected by subtraction, so you must factor first to reveal the common factor.

How do I know when I can cancel something?

You can cancel when the same factor appears in both the numerator and denominator. After factoring, we got $\frac{2x(5x-1)}{5x-1}$ , and (5x-1) appears in both places.

What if the denominator doesn't factor out completely?

That's fine! You can only cancel the parts that do factor out. Sometimes expressions don't simplify completely, and that's the final answer.

Do I need to worry about when x makes the denominator zero?

Yes! The original expression is undefined when $5x-1 = 0$ , so $x \neq \frac{1}{5}$ . Even after simplifying to 2x, this restriction still applies.

How can I check if my factoring is correct?

Multiply out your factored form! If $2x(5x-1)$ gives you back $10x^2-2x$ , then your factoring is correct.

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