Find the Equivalent Expression: Simplifying 15z²+50zx

Q: How do I know what the greatest common factor is?

Look at both the numbers and variables separately! For $ 15z^2 + 50zx $, find the GCF of 15 and 50 (which is 5), then find the highest power of each variable that appears in all terms (z appears once in both terms).

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

Break down each term completely! Write $ 15z^2 = 3 \cdot 5 \cdot z \cdot z $ and $ 50zx = 2 \cdot 5^2 \cdot z \cdot x $. Then pick out what's common to both: that's your GCF.

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

Always multiply it back out! Use the distributive property: $ 5z(3z + 10x) = 5z \cdot 3z + 5z \cdot 10x = 15z^2 + 50zx $. If you get the original expression, you're right!

Q: What if there's no common factor between terms?

Every pair of terms has at least a common factor of 1! But if there's truly no other common factor, then the expression is already in its simplest form and cannot be factored further.

Q: Why do we factor expressions anyway?

Factoring makes expressions easier to work with! It helps you solve equations, find zeros of functions, simplify fractions, and see patterns that aren't obvious in the expanded form.

Factoring Polynomials with Greatest Common Factor

Choose the expression that is equivalent to the following:

$15z^2+50zx$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find a common factor

00:05 Factorize 15 into factors 5 and 3

00:09 Factorize 50 into factors 5 and 10

00:19 Break down the square into products

00:23 Mark the common factors

00:48 Take out the common factors from the parentheses

00:55 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

Choose the expression that is equivalent to the following:

$15z^2+50zx$

Step-by-step solution

To solve the problem, the goal is to factor the expression $15z^2 + 50zx$ by finding the greatest common factor.

Step 1: Identify the greatest common factor (GCF):

The terms in the expression are $15z^2$ and $50zx$ .
The numerical coefficients are $15$ and $50$ , and their GCF is $5$ .
Both terms contain $z$ as a factor, so $z$ is also part of the GCF.
Therefore, the GCF of both terms is $5z$ .

Step 2: Factor the expression using the GCF:

Divide each term by the GCF $5z$ :
$\frac{15z^2}{5z} = 3z$
$\frac{50zx}{5z} = 10x$
Thus, the expression can be rewritten as the product of the GCF and the remaining terms: $5z(3z + 10x)$ .

Therefore, the expression $15z^2 + 50zx$ is equivalent to $5z(3z + 10x)$ .

Final Answer

$5z(3z+10x)$

Key Points to Remember

Essential concepts to master this topic

GCF Rule: Find the highest factor common to all terms
Technique: Divide each term by GCF: $\frac{15z^2}{5z} = 3z$ , $\frac{50zx}{5z} = 10x$
Check: Multiply factored form back: $5z(3z + 10x) = 15z^2 + 50zx$ ✓

Common Mistakes

Avoid these frequent errors

Finding only numerical GCF and ignoring variable factors
Don't just factor out 5 and write 5(3z² + 10zx) = wrong answer! This misses the common z factor and creates a more complex expression inside parentheses. Always check for both numerical AND variable factors in the GCF.

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 know what the greatest common factor is?

Look at both the numbers and variables separately! For $15z^2 + 50zx$ , find the GCF of 15 and 50 (which is 5), then find the highest power of each variable that appears in all terms (z appears once in both terms).

What if I can't see the GCF right away?

Break down each term completely! Write $15z^2 = 3 \cdot 5 \cdot z \cdot z$ and $50zx = 2 \cdot 5^2 \cdot z \cdot x$ . Then pick out what's common to both: that's your GCF.

How do I check if my factored answer is correct?

Always multiply it back out! Use the distributive property: $5z(3z + 10x) = 5z \cdot 3z + 5z \cdot 10x = 15z^2 + 50zx$ . If you get the original expression, you're right!

What if there's no common factor between terms?

Every pair of terms has at least a common factor of 1! But if there's truly no other common factor, then the expression is already in its simplest form and cannot be factored further.

Why do we factor expressions anyway?

Factoring makes expressions easier to work with! It helps you solve equations, find zeros of functions, simplify fractions, and see patterns that aren't obvious in the expanded form.

Can I factor out a negative GCF?

Yes! Sometimes factoring out a negative makes the expression cleaner. For example, $-6x - 12 = -6(x + 2)$ looks neater than $6(-x - 2)$ .

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