Find the Common Factor in 6x² + 8x: Algebraic Terms Practice

Q: Why isn't x the greatest common factor if both terms have x?

While both terms contain x, we need the greatest common factor! Since 6 and 8 share the factor 2, the GCF is $ 2x $, not just $ x $.

Q: How do I know which number factors are common?

Break down each coefficient completely: $ 6 = 2 \times 3 $ and $ 8 = 2 \times 4 $. The common numerical factor is 2.

Q: What if the terms had different powers of x?

Always take the lowest power that appears in all terms. For example, with $ 6x^3 + 8x $, the common variable factor would be $ x^1 $ (just $ x $).

Q: How can I check if I factored correctly?

Use the distributive property to expand your factored form. $ 2x(3x + 4) = 6x^2 + 8x $. If you get back the original expression, you're right!

Q: Is there ever more than one way to factor?

You can factor in steps, but there's only one greatest common factor. You might factor out 2 first, then x, but $ 2x $ is the most efficient single step.

Factoring Polynomials with Greatest Common Factor

We factored the expression

$6x^2+8x$

into its basic terms:

$6\cdot x\cdot x+8\cdot x$

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

$6x^2+8x$

into its basic terms:

$6\cdot x\cdot x+8\cdot x$

What common factor can be found in these terms?

Step-by-step solution

First, consider the expression $6x^2+8x$ . We want to factor out the greatest common factor of the terms.

Both terms, $6x^2$ and $8x$ , contain the factor $x$ . Therefore, $x$ is a common factor.

But we can keep factoring the numbers as well. $6$ can be factored to $2\cdot3$ , and $8$ can be factored to $2\cdot4$ .

Write each term showing the factor $x$ : $\blue2\cdot3\cdot \orange x\cdot x$ and $\blue 2\cdot4\cdot \orange x$ .

Therefore, the greatest common factor is $2\cdot x$ .

Final Answer

$2\cdot x$

Key Points to Remember

Essential concepts to master this topic

Rule: Find GCF by looking at both numerical and variable factors
Technique: Factor numbers separately: 6 = 2×3, 8 = 2×4, common is 2
Check: Factor out $2x$ to get $2x(3x + 4)$ ✓

Common Mistakes

Avoid these frequent errors

Only finding common numerical factors or only variable factors
Don't just factor out 2 or just x = incomplete factoring! This misses the greatest common factor and doesn't fully simplify the expression. Always find both the greatest common numerical factor AND the highest power of common variables together.

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 x the greatest common factor if both terms have x?

While both terms contain x, we need the greatest common factor! Since 6 and 8 share the factor 2, the GCF is $2x$ , not just $x$ .

How do I know which number factors are common?

Break down each coefficient completely: $6 = 2 \times 3$ and $8 = 2 \times 4$ . The common numerical factor is 2.

What if the terms had different powers of x?

Always take the lowest power that appears in all terms. For example, with $6x^3 + 8x$ , the common variable factor would be $x^1$ (just $x$ ).

How can I check if I factored correctly?

Use the distributive property to expand your factored form. $2x(3x + 4) = 6x^2 + 8x$ . If you get back the original expression, you're right!

Is there ever more than one way to factor?

You can factor in steps, but there's only one greatest common factor. You might factor out 2 first, then x, but $2x$ is the most efficient single step.

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