Find Equivalent Expressions for 2xy+x²+3x: Polynomial Simplification

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

Look at each term and find what appears in every single one. In $ 2xy + x^2 + 3x $, every term has an x, so x is the GCF!

Q: How do I check if my factored form is correct?

Use the distributive property to expand your answer back out. If you get the original expression, you're right! $ x(2y + x + 3) = 2xy + x^2 + 3x $ ✓

Q: What if some terms don't have the common factor?

Then those terms cannot be factored with the others. You can only factor out what appears in every single term of the expression.

Polynomial Factoring with Common Factors

Which of the expressions is equivalent to the expression?

$2xy+x^2+3x$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's begin by finding the common factor in the equation.

00:11 Now, break down the expression into smaller multiplications.

00:19 Great! Identify and mark those common factors. This step is crucial.

00:33 Next, let's extract the common factors from the parentheses.

00:38 And that's how we find the solution! Well done.

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?

$2xy+x^2+3x$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the greatest common factor (GCF) in the expression.
Step 2: Factor out the GCF from the expression.
Step 3: Compare the factored expression with the choices provided.

Now, let's work through each step:
Step 1: The expression given is $2xy + x^2 + 3x$ . The GCF of these terms is $x$ because it appears in each term.
Step 2: Factor out $x$ from each term, which gives: $x(2y + x + 3)$ This rewrites the expression in its factored form.
Step 3: Compare the factored form $x(2y + x + 3)$ with the answer choices.

Choice 1: $2x + 2y + 3$ does not match the factored form.
Choice 2: $x(2y + x + 3)$ exactly matches the factored form.
Choice 3: $2x(y + 4)$ does not match the factored form.
Choice 4: $2x(y + x + 3)$ does not match the factored form.

Therefore, the expression $2xy + x^2 + 3x$ is equivalent to $x(2y + x + 3)$ .

Final Answer

$x(2y+x+3)$

Key Points to Remember

Essential concepts to master this topic

Factoring: Find the greatest common factor (GCF) first
Technique: Factor out x from $2xy + x^2 + 3x$ to get $x(2y + x + 3)$
Check: Expand your factored form back to original expression ✓

Common Mistakes

Avoid these frequent errors

Not factoring out the complete GCF
Don't factor out just part of the common factor like factoring out 2 instead of x = incomplete factoring! This leaves terms that could still be simplified further. Always identify and factor out the complete greatest common factor from all terms.

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 and find what appears in every single one. In $2xy + x^2 + 3x$ , every term has an x, so x is the GCF!

What if I can't see a common factor?

Look carefully at the variables and their powers. Even if the numbers are different, variables like x, y, or z that appear in every term can be factored out.

How do I check if my factored form is correct?

Use the distributive property to expand your answer back out. If you get the original expression, you're right! $x(2y + x + 3) = 2xy + x^2 + 3x$ ✓

What if some terms don't have the common factor?

Then those terms cannot be factored with the others. You can only factor out what appears in every single term of the expression.

Can I factor out numbers and variables together?

Yes! If every term has the same number AND variable, factor both out. For example, $6x + 9x^2$ factors to $3x(2 + 3x)$ .

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