Reduce the Expression: Simplifying (16x^4-4x^3)/(2x)

Q: Why do I need to factor the numerator first?

Factoring helps you see common factors that can be cancelled! In this problem, factoring out $ 4x^3 $ from the numerator shows us how to simplify more easily.

Q: What's the rule for dividing variables with exponents?

When dividing variables with exponents, subtract the exponents: $ x^a ÷ x^b = x^{a-b} $. So $ x^4 ÷ x = x^3 $ and $ x^3 ÷ x = x^2 $.

Q: Can I divide each term in the numerator separately?

Yes, absolutely! When you have addition or subtraction in the numerator, you can divide each term individually: $ \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} $.

Q: What if there are restrictions on x?

Since we're dividing by $ 2x $, we need x ≠ 0 to avoid division by zero. This is an important restriction to remember!

Polynomial Division with Monomial Denominators

Reduce the following expression:

$\frac{16x^4-4x^3}{2x}$

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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 16 into factors 2 and 8

00:07 Let's break down power of 4 into factor to the power of 3 and multiplication by factor

00:12 Let's break down 4 into factors 2 and 2

00:16 Let's break down power of 3 into factor squared times the factor

00:21 Let's mark the common factors

00:45 Let's take out the common factors from the parentheses

01:02 Let's reduce what we can

01:07 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

Reduce the following expression:

$\frac{16x^4-4x^3}{2x}$

Step-by-step solution

Let's simplify the given expression:

$\frac{16x^4-4x^3}{2x}$ 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. We must first determine whether in the numerator we can factor out a common term. Following this we will proceed to reduce the possible expressions in the resulting fraction:

$\frac{16x^4-4x^3}{2x} \\ \frac{4x^3(4x-1)}{2x} \\ \frac{2x^2(4x-1)}{1}\\ \downarrow\\ \boxed{ 2x^2(4x-1)}$ Let's expand the parentheses in the resulting expression and we can therefore determine that the correct answer is answer a.

Final Answer

$8x^3-2x^2$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Factor common terms from numerator before dividing
Division Technique: Divide each term: $\frac{16x^4}{2x} = 8x^3$ , $\frac{-4x^3}{2x} = -2x^2$
Check Answer: Multiply result by denominator to get original numerator ✓

Common Mistakes

Avoid these frequent errors

Dividing only the coefficients and ignoring exponents
Don't just divide 16 ÷ 2 = 8 and 4 ÷ 2 = 2, getting 8x^4 - 2x^3! This ignores the division of variables. When dividing powers, subtract exponents: x^4 ÷ x = x^3 and x^3 ÷ x = x^2. Always divide both coefficients AND subtract exponents of like variables.

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 do I need to factor the numerator first?

Factoring helps you see common factors that can be cancelled! In this problem, factoring out $4x^3$ from the numerator shows us how to simplify more easily.

What's the rule for dividing variables with exponents?

When dividing variables with exponents, subtract the exponents: $x^a ÷ x^b = x^{a-b}$ . So $x^4 ÷ x = x^3$ and $x^3 ÷ x = x^2$ .

Can I divide each term in the numerator separately?

Yes, absolutely! When you have addition or subtraction in the numerator, you can divide each term individually: $\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}$ .

How do I check if my answer is correct?

Multiply your answer by the original denominator. If you get back the original numerator, you're right! For example: $(8x^3-2x^2) \times 2x = 16x^4-4x^3$ ✓

What if there are restrictions on x?

Since we're dividing by $2x$ , we need x ≠ 0 to avoid division by zero. This is an important restriction to remember!

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