Q. What is the factored form of \(3x + 24y\)?

Q: What is the correct factored form of 3x+24y?

3(x+8y). The greatest common factor is 3; factoring it out gives 3x+24y=3(x+8y).

Q: Why is 3xy(x+8y) incorrect?

It introduces extra factors x and y, changing the expression’s degree. Multiplying out gives 3x^2y+24xy, not 3x+24y.

Q: Why is 3(3x+24y) incorrect?

That multiplies the original by 3, giving 9x+72y. Factoring should produce an equivalent expression, not scale it.

Q: How can I check my factoring is correct?

Distribute the factor back: 3(x+8y) expands to 3x+24y. If expansion matches the original, factoring is correct.

Q: Is 3(x+8y) fully factored?

Yes over the integers: x+8y has no common numeric factor, so no further integer factoring is possible.

Q: What common mistakes should I avoid when factoring?

Don’t introduce or drop variables/factors (like extra xy), don’t multiply or divide all terms incorrectly, and always check by expanding to confirm you recover the original expression.

Answer

\[
3x+24y=3(x+8y)
\]

Detailed Explanation

We will factor the expression step by step.

Write the original expression:

\(3x + 24y\)
Find the greatest common factor (GCF) of the coefficients and variables:

The numerical coefficients are 3 and 24; their GCF is 3. The terms contain different variables (x and y), so there is no common variable factor. Therefore the overall common factor is 3.
Factor 3 out of each term by expressing each term as 3 times something:

\(3x = 3 \cdot x\)

\(24y = 3 \cdot 8y\)
Combine the factored parts into a single factored expression:

\(3x + 24y = 3\cdot x + 3\cdot 8y = 3(x + 8y)\)
Check the other given choices to confirm they are incorrect:
- \(3xy(x+8y)\) expands to \(3x^2y + 24xy^2\), which is not \(3x+24y\).
- \(3(3x+24y)\) equals \(9x+72y\), not the original expression.
- \(3xy(3x+24y)\) is even larger after expansion and does not match the original.
Conclusion — the correct factored form is:

\(3(x + 8y)\)

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FAQs

What is the correct factored form of \(3x+24y\)?

\(3(x+8y)\). The greatest common factor is 3; factoring it out gives \(3x+24y=3(x+8y)\).

Why is \(3xy(x+8y)\) incorrect?

It introduces extra factors \(x\) and \(y\), changing the expression’s degree. Multiplying out gives \(3x^2y+24xy\), not \(3x+24y\).

Why is \(3(3x+24y)\) incorrect?

That multiplies the original by 3, giving \(9x+72y\). Factoring should produce an equivalent expression, not scale it.

How can I check my factoring is correct?

Distribute the factor back: \(3(x+8y)\) expands to \(3x+24y\). If expansion matches the original, factoring is correct.

Is \(3(x+8y)\) fully factored?

Yes over the integers: \(x+8y\) has no common numeric factor, so no further integer factoring is possible.

Could I factor out a negative common factor instead?

What’s the general method to factor expressions like \(ax+by\)?

Find the greatest common factor (numeric and variable parts) of the terms and divide each term by it, then place the GCF outside parentheses and the simplified terms inside.

What common mistakes should I avoid when factoring?

Don’t introduce or drop variables/factors (like extra \(xy\)), don’t multiply or divide all terms incorrectly, and always check by expanding to confirm you recover the original expression.

Factor out the greatest common factor
Correct answer: 3(x + 8y).

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