Factor Out Common Terms in 4x²+2x: Algebraic Simplification

Q: How do I find the greatest common factor (GCF)?

Look at each term separately! In $ 4x^2 + 2x $, the first term has factors 2·2·x·x and the second has 2·x. The GCF is what appears in both terms: $ 2x $.

Q: Why can't I just factor out x?

You could, but that's not completely factored! If you factor out just x, you get $ x(4x + 2) $, but notice that the terms inside still share a factor of 2. Always factor out the greatest common factor.

Q: How do I check if my factoring is correct?

Use the distributive property to expand your answer! For $ 2x(2x + 1) $, multiply: $ 2x \cdot 2x + 2x \cdot 1 = 4x^2 + 2x $. If it matches the original, you're right!

Q: What if there's no common factor?

Sometimes polynomials are already in their simplest form! For example, $ 3x^2 + 5y $ has no common factors, so it can't be factored further.

Q: Can the common factor include numbers and variables?

Absolutely! The GCF can be a combination of numbers and variables. In our example, $ 2x $ includes both the number 2 and the variable x.

Factoring Polynomials with Common Factors

We factored the expression

$4x^2+2x$ into its basic terms:

$2\cdot2\cdot x\cdot x+2\cdot x$

Take out the common factor from the factored expression

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

We factored the expression

$4x^2+2x$ into its basic terms:

$2\cdot2\cdot x\cdot x+2\cdot x$

Take out the common factor from the factored expression

Step-by-step solution

To factor the expression $4x^2 + 2x$ , first notice that each term shares a common factor of $2x$ : $\blue 2\cdot2\cdot \orange x\cdot x+\blue 2\cdot \orange x$

Start by factoring out $2x$ :

$4x^2 + 2x = 2x(2x + 1)$

Thus, the factored expression is $2x(2x + 1)$ , as $2 \cdot x$ is the common factor.

Final Answer

$2x(2x + 1)$

Key Points to Remember

Essential concepts to master this topic

Rule: Identify the greatest common factor from all terms
Technique: Factor $4x^2 + 2x$ as $2x(2x + 1)$
Check: Expand $2x(2x + 1) = 4x^2 + 2x$ matches original ✓

Common Mistakes

Avoid these frequent errors

Factoring out only part of the common factor
Don't factor out just x from 4x² + 2x = x(4x + 2)! This misses the common factor of 2, leaving the expression partially unfactored. Always find the greatest common factor (GCF) which includes all shared variables and numbers.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 4x^2 + 6x $$

FAQ

Everything you need to know about this question

How do I find the greatest common factor (GCF)?

Look at each term separately! In $4x^2 + 2x$ , the first term has factors 2·2·x·x and the second has 2·x. The GCF is what appears in both terms: $2x$ .

Why can't I just factor out x?

You could, but that's not completely factored! If you factor out just x, you get $x(4x + 2)$ , but notice that the terms inside still share a factor of 2. Always factor out the greatest common factor.

How do I check if my factoring is correct?

Use the distributive property to expand your answer! For $2x(2x + 1)$ , multiply: $2x \cdot 2x + 2x \cdot 1 = 4x^2 + 2x$ . If it matches the original, you're right!

What if there's no common factor?

Sometimes polynomials are already in their simplest form! For example, $3x^2 + 5y$ has no common factors, so it can't be factored further.

Can the common factor include numbers and variables?

Absolutely! The GCF can be a combination of numbers and variables. In our example, $2x$ includes both the number 2 and the variable x.

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