Find the Common Factor in 2x²+7x: Breaking Down Terms

Q: Why isn't 2x the common factor if it's in the first term?

Look carefully! The first term $ 2x^2 $ contains two x's, but the second term $ 7x $ only has one x. Since 2x isn't in both terms, it can't be the common factor.

Q: How do I know x is really in both terms?

Break each term into its basic factors: $ 2x^2 = 2 \cdot x \cdot x $ and $ 7x = 7 \cdot x $. You can see that x appears in both broken-down forms!

Q: Can I factor out numbers and variables separately?

No! You must factor out the complete common factor all at once. In this case, since only x appears in both terms, x is the only common factor.

Q: How do I check my factoring is correct?

Use the distributive property to multiply back: $ x(2x + 7) = x \cdot 2x + x \cdot 7 = 2x^2 + 7x $. If you get the original expression, you're right!

Factoring Polynomials with Common Variables

The expression $2x^2+7x$ is broken down into basic terms:

$2\cdot x\cdot x+7\cdot x$

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

The expression $2x^2+7x$ is broken down into basic terms:

$2\cdot x\cdot x+7\cdot x$

What common factor can be found in these terms?

Step-by-step solution

To solve this problem, we need to find the common factor in the expression $2x^2 + 7x$ , which is given as broken down into basic forms: $2 \cdot x \cdot x + 7 \cdot x$ .

Let's go through the solution step-by-step:

Step 1: Examine each term in the expression.
The first term is $2 \cdot x \cdot x$ , equivalent to $2x^2$ .
The second term is $7 \cdot x$ .
Step 2: Identify factors in each term.
For $2x^2$ , we have the factors $2$ , $x$ , and another $x$ .
For $7x$ , the factors are $7$ and $x$ .
Step 3: Determine the common factor across all terms.
The common factor present in both terms is $x$ .

After identifying the shared factor $x$ , it can be factored out of both terms in the expression
Factorization: $2x^2 + 7x = x(2x + 7)$ .

Therefore, the common factor between the terms is $x$ .

Final Answer

$x$

Key Points to Remember

Essential concepts to master this topic

Rule: Find factors that appear in every term of the expression
Technique: Break down $2x^2 + 7x$ into $2 \cdot x \cdot x + 7 \cdot x$
Check: Multiply factored form $x(2x + 7)$ back to get original expression ✓

Common Mistakes

Avoid these frequent errors

Choosing the largest coefficient as the common factor
Don't pick 7 as the common factor just because it's the largest number = wrong factorization! The number 7 only appears in one term, not both. Always identify factors that exist in every single term of the expression.

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 2x the common factor if it's in the first term?

Look carefully! The first term $2x^2$ contains two x's, but the second term $7x$ only has one x. Since 2x isn't in both terms, it can't be the common factor.

How do I know x is really in both terms?

Break each term into its basic factors: $2x^2 = 2 \cdot x \cdot x$ and $7x = 7 \cdot x$ . You can see that x appears in both broken-down forms!

What if there's no common factor?

If terms share no common factors, the expression is already in its simplest form. But most factoring problems are designed to have at least one common factor, so double-check your work!

Can I factor out numbers and variables separately?

No! You must factor out the complete common factor all at once. In this case, since only x appears in both terms, x is the only common factor.

How do I check my factoring is correct?

Use the distributive property to multiply back: $x(2x + 7) = x \cdot 2x + x \cdot 7 = 2x^2 + 7x$ . If you get the original expression, you're right!

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