Simplify the Polynomial Fraction: (16x-4x²)/(4-x)

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

You can only cancel common factors, not individual terms! The expression $ 16x-4x^2 $ is a sum, so you must factor it completely first before canceling anything.

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

Look for the greatest common factor (GCF) of all terms. In $ 16x-4x^2 $, both terms have 4x as a factor, so factor out 4x: $ 4x(4-x) $

Q: What happens to the negative sign in (4-x)?

Notice that $ 4-x = -(x-4) $. When you factor out the negative, you get $ 4x \cdot -(x-4) $, which simplifies nicely with the denominator!

Q: Can this fraction be simplified differently?

No! There's only one correct simplified form: $ 4x $. Any other approach that doesn't follow proper factorization rules will give you the wrong answer.

Polynomial Fraction Reduction with Common Factors

Simplify the following expression:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:04 Let's break down 16 into factors 4 and 4

00:10 Let's break down the power into multiplications

00:17 Let's mark the common factors

00:46 Let's take out the common factors from the parentheses

00:54 Let's reduce what we can

01:02 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:

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

Step-by-step solution

Let's simplify the given expression:

$\frac{16x-4x^2}{4-x}$ Remember that we can reduce complete expressions only when both the numerator and denominator are completely factored into multiplication expressions,

We will use factorization, identify that in the numerator we can factor out a common term, then reduce the expressions in the obtained fraction(reduction sign):

$\frac{16x-4x^2}{4-x} \\ \frac{4x(x-4)}{x-4} \\ \downarrow\\ \boxed{4x}$ Therefore, the correct answer is answer B.

Final Answer

$4x$

Key Points to Remember

Essential concepts to master this topic

Factorization Rule: Both numerator and denominator must be completely factored
Factor Technique: From 16x-4x², extract 4x to get 4x(4-x)
Verification Check: Substitute x=2: 4(2)=8 matches original expression ✓

Common Mistakes

Avoid these frequent errors

Canceling terms before complete factorization
Don't cancel individual terms like 16x with 4 = wrong simplification! This ignores proper algebraic rules and gives incorrect results. Always factor completely first, then reduce common factors from numerator and denominator.

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 cancel the x terms directly?

You can only cancel common factors, not individual terms! The expression $16x-4x^2$ is a sum, so you must factor it completely first before canceling anything.

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

Look for the greatest common factor (GCF) of all terms. In $16x-4x^2$ , both terms have 4x as a factor, so factor out 4x: $4x(4-x)$

What happens to the negative sign in (4-x)?

Notice that $4-x = -(x-4)$ . When you factor out the negative, you get $4x \cdot -(x-4)$ , which simplifies nicely with the denominator!

Can this fraction be simplified differently?

No! There's only one correct simplified form: $4x$ . Any other approach that doesn't follow proper factorization rules will give you the wrong answer.

How can I check if 4x is the right answer?

Pick any value for x (except x=4 to avoid division by zero) and substitute into both the original expression and your answer. If they're equal, you're correct!

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