Factoring 3x² + 12x: Finding the Common Factor Step-by-Step

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

Look at both the coefficients and variables separately. For $ 3x^2 + 12x $: coefficients 3 and 12 have GCF of 3, variables $ x^2 $ and x have GCF of x. Combined GCF is 3x.

Q: Why can't I factor out just 3 or just x?

You could factor out just 3 or just x, but that's not the greatest common factor! Since both terms contain 3x, factoring out the complete GCF gives you the simplest form.

Q: How do I know what's left inside the parentheses?

Divide each original term by the GCF. For $ 3x^2 ÷ 3x = x $ and $ 12x ÷ 3x = 4 $, so you get $ 3x(x + 4) $.

Q: Can I check my factoring without multiplying everything out?

Yes! Pick a simple value like x = 1. Check: $ 3(1)^2 + 12(1) = 15 $ and $ 3(1)(1 + 4) = 15 $. If they match, your factoring is likely correct!

Factoring Polynomials with Greatest Common Factors

We factored the expression $3x^2 + 12x$ into its basic terms:

$3 \cdot x \cdot x + 12 \cdot x$

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 $3x^2 + 12x$ into its basic terms:

$3 \cdot x \cdot x + 12 \cdot x$

Take out the common factor from the factored expression

Step-by-step solution

To factor the expression $3x^2 + 12x$ , we start by looking for the greatest common factor (GCF) of the terms $3x^2$ and $12x$ . The GCF is $3x$ . We can factor out $3x$ from each term:

$3x^2+12x=\blue3\cdot \orange x\cdot x+\blue 3\cdot4\cdot \orange x$ .

This allows us to write the expression as $3x(x + 4)$ .

Final Answer

$3x \left(x + 4 \right)$

Key Points to Remember

Essential concepts to master this topic

Rule: Find the GCF of all terms before factoring
Technique: Factor out 3x from 3x² + 12x = 3x(x + 4)
Check: Multiply back: 3x(x + 4) = 3x² + 12x ✓

Common Mistakes

Avoid these frequent errors

Factoring out only numerical coefficients
Don't factor out just 3 from 3x² + 12x = 3(x² + 4x)! This leaves x unfactored in the second term. Always factor out the complete GCF including variables: 3x(x + 4).

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 both the coefficients and variables separately. For $3x^2 + 12x$ : coefficients 3 and 12 have GCF of 3, variables $x^2$ and x have GCF of x. Combined GCF is 3x.

Why can't I factor out just 3 or just x?

You could factor out just 3 or just x, but that's not the greatest common factor! Since both terms contain 3x, factoring out the complete GCF gives you the simplest form.

What if the terms don't seem to have any common factors?

Look more carefully! Even if coefficients are different, check if variables have common factors. For example, $2x^3 + 5x$ has GCF of x, giving you $x(2x^2 + 5)$ .

How do I know what's left inside the parentheses?

Divide each original term by the GCF. For $3x^2 ÷ 3x = x$ and $12x ÷ 3x = 4$ , so you get $3x(x + 4)$ .

Can I check my factoring without multiplying everything out?

Yes! Pick a simple value like x = 1. Check: $3(1)^2 + 12(1) = 15$ and $3(1)(1 + 4) = 15$ . If they match, your factoring is likely correct!

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