Simplify the Algebraic Fraction: (3x-1)/(15x²-5x²)

Fraction Simplification with Common Factor Cancellation

Simplify:

3x115x25x2 \frac{3x-1}{15x^2-5x^2}

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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 3
00:13 Let's break down the power of 3 into square times the factor
00:22 Let's mark the common factors
00:42 Let's take out the common factors from the parentheses
00:54 Let's reduce what we can
01:00 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Simplify:

3x115x25x2 \frac{3x-1}{15x^2-5x^2}

2

Step-by-step solution

Let's simplify the given expression:

3x115x25x2 \frac{3x-1}{15x^2-5x^2} Remember that we can reduce complete expressions only when both the numerator and denominator are completely factored into multiplication expressions,

For this, we'll use factorization, identify where in the denominator we can factor out a common factor, do this, then reduce the expressions possible in the fraction we get (reduction sign):

3x115x35x23x15x2(3x1)15x2 \frac{3x-1}{15x^3-5x^2} \\ \frac{3x-1}{5x^2(3x-1)} \\ \downarrow\\ \boxed{\frac{1}{5x^2}} In the first stage, to factor out the common factor in the denominator, we also used the law of exponents:aman=am+n a^m\cdot a^n=a^{m+n}

Therefore, the correct answer is answer B.

3

Final Answer

15x2 \frac{1}{5x^2}

Key Points to Remember

Essential concepts to master this topic
  • Rule: Factor completely before canceling any terms from fractions
  • Technique: Factor out 5x² from denominator: 15x²-5x² = 10x²
  • Check: Multiply result by original denominator to get numerator: 15x25x2(3x1)=3x1 \frac{1}{5x^2} \cdot 5x^2(3x-1) = 3x-1

Common Mistakes

Avoid these frequent errors
  • Canceling terms without complete factorization
    Don't cancel 3x from numerator with parts of 15x² or 5x² = wrong answer! This breaks fraction rules because you need complete factorization first. Always factor the entire denominator completely, then look for common factors to cancel.

Practice Quiz

Test your knowledge with interactive questions

Determine if the simplification shown below is correct:

\( \frac{7}{7\cdot8}=8 \)

FAQ

Everything you need to know about this question

Why can't I just subtract 15x² - 5x² = 10x² in the denominator?

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You absolutely should! That's the first step. Combine like terms to get 3x110x2 \frac{3x-1}{10x^2} . But this doesn't match any answer choice, so there might be a typo in the original problem.

The explanation shows 15x³ - 5x² in the denominator. Is that a typo?

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Yes, that appears to be a typo! The explanation factors 15x35x2=5x2(3x1) 15x^3-5x^2 = 5x^2(3x-1) , which would give the correct answer 15x2 \frac{1}{5x^2} .

How do I know when I can cancel factors?

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You can only cancel factors (terms connected by multiplication) that appear in both the numerator and denominator. Never cancel terms that are added or subtracted!

What if the numerator and denominator have no common factors?

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Then the fraction is already in simplest form! Not every fraction can be reduced further. Always check for common factors first by factoring completely.

Should I always factor out the GCF first?

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Yes! Start by finding the Greatest Common Factor in each polynomial. This makes it easier to spot what can be canceled between numerator and denominator.

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