Simplify the Rational Expression: (5x²-1)/(15x⁴-3x²)

Q: Why can't I just divide the coefficients and subtract the exponents?

You can only do this when expressions are multiplied together, not when they're added or subtracted! Since we have $ 5x^2-1 $ (subtraction), we must factor first before simplifying.

Q: How do I know what to factor out from the denominator?

Look for the greatest common factor (GCF) of all terms. Here, both $ 15x^4 $ and $ 3x^2 $ are divisible by $ 3x^2 $, so factor that out first.

Q: What if the numerator doesn't factor nicely?

That's okay! In this problem, $ 5x^2-1 $ doesn't factor further with real numbers. The key is factoring the denominator to reveal the common factor $ (5x^2-1) $ that cancels.

Q: Can I cancel the x² terms directly?

No! You can only cancel complete factors, not individual terms within a sum or difference. Always factor first, then look for identical factors in numerator and denominator to cancel.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:04 Let's factorize 15 into factors 5 and 3

00:10 Let's break down power of 4 into 2 squared factors

00:20 Let's mark the common factors

00:41 Let's take out the common factors from the parentheses

00:53 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:

$\frac{5x^2-1}{15x^4-3x^2}$

2

Step-by-step solution

Let's simplify the given expression:

$\frac{5x^2-1}{15x^4-3x^2}$ 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 that in the denominator we can factor out a common term, do this, then reduce the expressions possible in the fraction we got (reduction sign):

$\frac{5x^2-1}{15x^4-3x^2} \\ \frac{5x^2-1}{3x^2(5x^2-1)} \\ \downarrow\\ \boxed{\frac{1}{3x^2}}$ In the first stage, to factor out the common term in the denominator, we also used the law of exponents: $a^m\cdot a^n=a^{m+n}$

Therefore, the correct answer is answer B.

3

Final Answer

$\frac{1}{3x^2}$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Both numerator and denominator must be completely factored first
Technique: Factor out $3x^2$ from denominator: $15x^4-3x^2 = 3x^2(5x^2-1)$
Check: Multiply simplified result by denominator should give original numerator ✓

Common Mistakes

Avoid these frequent errors

Trying to cancel terms instead of factors
Don't cancel individual terms like the 5x² in numerator with 15x⁴ in denominator = incorrect simplification! Terms can only be reduced when they are complete factors of both numerator and denominator. Always factor completely first, then cancel identical factors.

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 divide the coefficients and subtract the exponents?

+

You can only do this when expressions are multiplied together, not when they're added or subtracted! Since we have $5x^2-1$ (subtraction), we must factor first before simplifying.

How do I know what to factor out from the denominator?

+

Look for the greatest common factor (GCF) of all terms. Here, both $15x^4$ and $3x^2$ are divisible by $3x^2$ , so factor that out first.

What if the numerator doesn't factor nicely?

+

That's okay! In this problem, $5x^2-1$ doesn't factor further with real numbers. The key is factoring the denominator to reveal the common factor $(5x^2-1)$ that cancels.

Can I cancel the x² terms directly?

+

No! You can only cancel complete factors, not individual terms within a sum or difference. Always factor first, then look for identical factors in numerator and denominator to cancel.

How do I check if my simplified answer is correct?

+

Multiply your answer by the original denominator. You should get the original numerator: $\frac{1}{3x^2} \times (15x^4-3x^2) = 5x^2-1$ ✓

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Simplify the Rational Expression: (5x²-1)/(15x⁴-3x²)

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