Simplify the Algebraic Fraction: (x³-16x²)/(8x) Step by Step

Q: Why can't I just divide each term in the numerator by 8x separately?

While you can split fractions for addition/subtraction, it's not the simplest form. Factoring first gives you $ \frac{x^2(x-16)}{8} $, which is cleaner than $ \frac{x^2}{8} - 2x $.

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

Look for the greatest common factor (GCF) of all terms. Here, both $ x^3 $ and $ 16x^2 $ contain $ x^2 $, so factor out $ x^2 $.

Q: What if there's no common factor to cancel?

Then the fraction is already in simplest form! Not every algebraic fraction can be simplified further. Always check for common factors first though.

Q: Can I cancel the x's before factoring?

No! You can only cancel factors that multiply the entire numerator and denominator. Since the numerator has subtraction, you must factor first to create multiplication.

Q: Why does the answer have x to the first power instead of x squared?

After factoring out $ x^2 $ and canceling with $ x $ in the denominator, we get $ x^{2-1} = x^1 = x $. This uses the exponent rule $ \frac{a^m}{a^n} = a^{m-n} $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's get started by breaking it down step by step.

00:13 We'll break 3 raised to a power into a product of a factor, and then factor squared.

00:23 Next, we'll identify and mark any common factors. Nice work!

00:42 Now, let's factor out the common terms from inside the parentheses.

00:50 Great job! Let's simplify the expression by reducing what we can.

00:57 This solution works well. Let's keep going to explore more.

01:06 We'll break the power into smaller products. Keep it up!

01:18 Let's find and mark those common factors again.

01:27 We're doing great! Factor out the terms from the parentheses.

01:31 And there you have it! This is another solution to the problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Simplify:

$\frac{x^3-16x^2}{8x}$

2

Step-by-step solution

Let's simplify the given expression:

$\frac{x^3-16x^2}{8x}$ 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 numerator we can factor out a common term, do this, then we'll use the exponent rule mentioned later:

$\frac{x^3-16x^2}{8x} \\ \frac{x^2(x-16)}{8x} \\ \frac{x^{2}(x-16)}{8x^1} \\ \frac{x^{2-1}(x-16)}{8} \\ \downarrow\\ \boxed{\frac{x(x-16)}{8} }$

In the final steps, instead of using the reduction sign, we used the exponent law:

$\frac{a^m}{a^n}=a^{m-n}=\frac{a^{m-n}}{1}$ (Actually, reduction is simply the quick way to apply the mentioned exponent law, but of course - the operations are identical).

From opening the parentheses in the fraction's numerator that we got, we can identify that the correct (best) answer is answer B.

3

Final Answer

$\frac{x^2-16x}{8}$

Key Points to Remember

Essential concepts to master this topic

Factorization Rule: Factor out greatest common factor from numerator first
Technique: Factor $x^3-16x^2 = x^2(x-16)$ before dividing
Check: Multiply result by denominator: $\frac{x^2(x-16)}{8} \cdot 8x = x^3-16x^2$ ✓

Common Mistakes

Avoid these frequent errors

Dividing terms separately without factoring
Don't divide $\frac{x^3}{8x} - \frac{16x^2}{8x}$ = $\frac{x^2}{8} - 2x$ ! This creates two separate fractions instead of one simplified expression. Always factor the numerator completely first, then cancel common factors as a single unit.

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 each term in the numerator by 8x separately?

+

While you can split fractions for addition/subtraction, it's not the simplest form. Factoring first gives you $\frac{x^2(x-16)}{8}$ , which is cleaner than $\frac{x^2}{8} - 2x$ .

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

+

Look for the greatest common factor (GCF) of all terms. Here, both $x^3$ and $16x^2$ contain $x^2$ , so factor out $x^2$ .

What if there's no common factor to cancel?

+

Then the fraction is already in simplest form! Not every algebraic fraction can be simplified further. Always check for common factors first though.

Can I cancel the x's before factoring?

+

No! You can only cancel factors that multiply the entire numerator and denominator. Since the numerator has subtraction, you must factor first to create multiplication.

Why does the answer have x to the first power instead of x squared?

+

After factoring out $x^2$ and canceling with $x$ in the denominator, we get $x^{2-1} = x^1 = x$ . This uses the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ .

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