Find the Common Factor in z²+9z: Basic Term Analysis

Q: Why isn't 9 the common factor if it's in one of the terms?

A common factor must appear in every term! Since 9 doesn't divide evenly into $ z^2 $, it can't be factored out from the entire expression.

Q: Could 3z be a common factor instead of just z?

No, because 3 doesn't appear in the first term $ z^2 $. While $ 9z = 3 \cdot 3z $, we can't write $ z^2 $ as $ 3 \cdot (something) $ with integer coefficients.

Q: How do I know I found the greatest common factor?

The greatest common factor (GCF) is the largest expression that divides all terms. Here, $ z $ is the highest power of the variable that appears in every term.

Q: How do I check if my factoring is correct?

Use the distributive property to expand your factored form. If you get back to the original expression, your factoring is correct!

Factoring Polynomials with Common Variables

We factored the expression

$z^2+9z$ into its basic terms:

$z\cdot z+9\cdot z$

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

$z^2+9z$ into its basic terms:

$z\cdot z+9\cdot z$

What common factor can be found in these terms?

Step-by-step solution

To factor the expression $z^2+9z$ , we write each term as a product of factors:

$z^2=z\cdot \orange z$ and $9z = 9 \cdot \orange z$ .

We observe that both terms have a common factor of $z$ , so the greatest factor is $z$ .

Final Answer

$z$

Key Points to Remember

Essential concepts to master this topic

Common Factor Rule: Look for variables or numbers present in every term
Technique: Write each term as products: $z^2 = z \cdot z$ and $9z = 9 \cdot z$
Check: Factor out and expand: $z(z + 9) = z^2 + 9z$ ✓

Common Mistakes

Avoid these frequent errors

Choosing the largest coefficient as the common factor
Don't pick 9 just because it's the biggest number = wrong factorization! The number 9 doesn't appear in the first term $z^2$ , so it can't be factored out. Always find factors that appear in every single term.

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 9 the common factor if it's in one of the terms?

A common factor must appear in every term! Since 9 doesn't divide evenly into $z^2$ , it can't be factored out from the entire expression.

Could 3z be a common factor instead of just z?

No, because 3 doesn't appear in the first term $z^2$ . While $9z = 3 \cdot 3z$ , we can't write $z^2$ as $3 \cdot (something)$ with integer coefficients.

How do I know I found the greatest common factor?

The greatest common factor (GCF) is the largest expression that divides all terms. Here, $z$ is the highest power of the variable that appears in every term.

What if there was no common factor?

If no common factor exists, the expression is already in its simplest form. But always check carefully - even a simple variable like $x$ or a number like 2 can be a common factor!

How do I check if my factoring is correct?

Use the distributive property to expand your factored form. If you get back to the original expression, your factoring is correct!

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