Find the Common Factor in 3x²+9x: Breaking Down Terms

Q: Why isn't just 3 or just x the correct answer?

While both $ 3 $ and $ x $ are common factors, we need the greatest common factor. Since both terms contain both 3 and x, the GCF is $ 3x $.

Q: How do I know I found the greatest common factor?

Break each term into its prime factors. For $ 3x^2 + 9x $, that's $ 3 \cdot x \cdot x + 3 \cdot 3 \cdot x $. The GCF includes every factor that appears in all terms.

Q: What if I factored out 3x and got the wrong expression?

Check your work by expanding! If you factored correctly, $ 3x(x + 3) = 3x^2 + 9x $. If expanding doesn't give you the original expression, try again.

Q: Can I factor out 9x instead?

No! The first term $ 3x^2 $ doesn't contain the factor 9, so 9x cannot be factored from both terms. Only factors present in every term can be factored out.

Q: Why do we want to find the greatest common factor?

Finding the GCF completely factors the expression in one step. This makes it easier to solve equations, simplify fractions, and work with the expression in future problems.

Factoring Polynomials with Greatest Common Factor

We factored the expression

$3x^2+9x$ into its basic terms:

$3\cdot x\cdot x+9\cdot x$

What common factor can be found in these terms?

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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

$3x^2+9x$ into its basic terms:

$3\cdot x\cdot x+9\cdot x$

What common factor can be found in these terms?

Step-by-step solution

First, consider the expression $3x^2+9x$ . We want to factor out the greatest common factor of the terms.

Both terms, $3x^2$ and $9x$ , contain the factor $x$ . Therefore, $x$ is a common factor.

Write each term showing the factor $x$ : $3\cdot x\cdot\orange x$ and $9\cdot\orange x$ .

However, we can further more factor the number $9$ , to $3\cdot3$ .

So, we can see there's another common factor, $3$ , as we can see in both terms: $\blue3\cdot x\cdot x$ and $\blue3\cdot3\cdot x$ .

$\blue3\cdot x\cdot\orange x$ and $\blue 3\cdot3\cdot \orange x$

The greatest common factor is then $3x$ .

Final Answer

$3x$

Key Points to Remember

Essential concepts to master this topic

Rule: Identify all factors common to every term in the expression
Technique: Break down $3x^2 + 9x$ into $3 \cdot x \cdot x + 3 \cdot 3 \cdot x$
Check: Factor out $3x$ to get $3x(x + 3)$ , then expand back ✓

Common Mistakes

Avoid these frequent errors

Finding only one common factor instead of the greatest
Don't stop at just finding $3$ or just $x$ = incomplete factoring! This leaves more factoring work undone and doesn't simplify the expression fully. Always find the greatest common factor by identifying ALL factors that appear in every term.

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

Why isn't just 3 or just x the correct answer?

While both $3$ and $x$ are common factors, we need the greatest common factor. Since both terms contain both 3 and x, the GCF is $3x$ .

How do I know I found the greatest common factor?

Break each term into its prime factors. For $3x^2 + 9x$ , that's $3 \cdot x \cdot x + 3 \cdot 3 \cdot x$ . The GCF includes every factor that appears in all terms.

What if I factored out 3x and got the wrong expression?

Check your work by expanding! If you factored correctly, $3x(x + 3) = 3x^2 + 9x$ . If expanding doesn't give you the original expression, try again.

Can I factor out 9x instead?

No! The first term $3x^2$ doesn't contain the factor 9, so 9x cannot be factored from both terms. Only factors present in every term can be factored out.

Why do we want to find the greatest common factor?

Finding the GCF completely factors the expression in one step. This makes it easier to solve equations, simplify fractions, and work with the expression in future problems.

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