Find Expressions Equivalent to 8x²+4x: Polynomial Practice

Q: How do I find the GCF of coefficients and variables together?

Find the GCF of numbers separately, then the lowest power of each variable. For $ 8x^2 + 4x $: GCF of 8 and 4 is 4, lowest power of x is x¹, so GCF = 4x.

Q: What if I can't see the common factor right away?

List the prime factorization of each term! For example: $ 8x^2 = 2^3 \cdot x^2 $ and $ 4x = 2^2 \cdot x $. The common factors are $ 2^2 \cdot x = 4x $.

Q: How do I know my factoring is correct?

Always distribute back to check! Multiply $ 4x(2x + 1) = 4x \cdot 2x + 4x \cdot 1 = 8x^2 + 4x $. If you get the original expression, you're right!

Q: Can I factor this expression differently?

You could factor out smaller amounts like $ 2x(4x + 2) $, but it's not fully factored. Always factor out the greatest common factor to get the simplest form.

Q: What if one of the answer choices looks close but not exact?

Be careful! Expand each choice to double-check. For example, $ 4x(2x + 6) = 8x^2 + 24x $ which is NOT equal to $ 8x^2 + 4x $.

Polynomial Factoring with Greatest Common Factor

Which of the expressions is equivalent to the expression?

$8x^2+4x$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Take out common factor

00:03 Break down 8 into factors 4 and 2

00:07 Break down the square into products

00:13 Mark the common factors

00:24 Extract the common factors from the parentheses

00:30 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

Which of the expressions is equivalent to the expression?

$8x^2+4x$

Step-by-step solution

Let's solve the problem step by step:

Step 1: Identify the greatest common factor (GCF) of the coefficients in the expression $8x^2 + 4x$ . The numbers are 8 and 4, and the GCF is 4.
Step 2: Factor out the common variable. Both terms have $x$ as a common variable factor, so the GCF of the variable part is $x$ .
Step 3: Factor the expression using the GCF. We take $4x$ as a common factor from both terms:
$8x^2$ can be rewritten as $4x \cdot 2x$ .
$+4x$ can be rewritten as $4x \cdot 1$ .
Step 4: Write the factored expression:
$8x^2 + 4x = 4x(2x + 1)$ .
Step 5: Verify by checking each option. The expression we obtained $4x(2x + 1)$ matches the choice with 2.

Therefore, the equivalent expression is $4x(2x+1)$ .

Final Answer

$4x(2x+1)$

Key Points to Remember

Essential concepts to master this topic

GCF Rule: Find greatest common factor of coefficients and variables
Technique: Factor out 4x from $8x^2 + 4x = 4x(2x + 1)$
Check: Distribute back: 4x(2x + 1) = 8x² + 4x ✓

Common Mistakes

Avoid these frequent errors

Factoring out only part of the GCF
Don't factor out just 4 and leave x behind = incomplete factoring like 4(2x² + x)! This misses the common x factor and doesn't fully simplify. Always factor out the complete GCF including all common variables.

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 GCF of coefficients and variables together?

Find the GCF of numbers separately, then the lowest power of each variable. For $8x^2 + 4x$ : GCF of 8 and 4 is 4, lowest power of x is x¹, so GCF = 4x.

What if I can't see the common factor right away?

List the prime factorization of each term! For example: $8x^2 = 2^3 \cdot x^2$ and $4x = 2^2 \cdot x$ . The common factors are $2^2 \cdot x = 4x$ .

How do I know my factoring is correct?

Always distribute back to check! Multiply $4x(2x + 1) = 4x \cdot 2x + 4x \cdot 1 = 8x^2 + 4x$ . If you get the original expression, you're right!

Can I factor this expression differently?

You could factor out smaller amounts like $2x(4x + 2)$ , but it's not fully factored. Always factor out the greatest common factor to get the simplest form.

What if one of the answer choices looks close but not exact?

Be careful! Expand each choice to double-check. For example, $4x(2x + 6) = 8x^2 + 24x$ which is NOT equal to $8x^2 + 4x$ .

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