factor using GCF with solution 10x²y³+20xy+5

To factor the expression \(10x^2y^3 + 20xy + 5\) using the Greatest Common Factor (GCF), follow these steps:

### Step 1: Identify the GCF of the terms.
- The terms are \(10x^2y^3\), \(20xy\), and \(5\).
- **Numerical coefficients:** The GCF of the numbers \(10\), \(20\), and \(5\) is \(5\).
- **Variable part:** The GCF of the variables \(x^2y^3\), \(xy\), and \(5\) (which has no variables) is simply \(1\), as there are no common variables in all terms.

So, the GCF of the entire expression is \(5\).

### Step 2: Factor out the GCF.
- Divide each term by the GCF and write the expression as a product of the GCF and the resulting expression.

\[
\begin{aligned}
10x^2y^3 + 20xy + 5 &= 5 \left(\frac{10x^2y^3}{5} + \frac{20xy}{5} + \frac{5}{5}\right) \\
&= 5 \left(2x^2y^3 + 4xy + 1\right)
\end{aligned}
\]

### Final Answer:
\[
10x^2y^3 + 20xy + 5 = 5(2x^2y^3 + 4xy + 1)
\]

So, the factored form of \(10x^2y^3 + 20xy + 5\) is \(5(2x^2y^3 + 4xy + 1)\).

Would you like further details or have any questions?

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Here are 8 related questions to expand your understanding:

1. How would you factor \(15x^3y + 25x^2 - 10xy\) using the GCF method?
2. What happens if the GCF includes variables, like in \(6x^2y - 9xy^2\)?
3. Can you explain how to factor a polynomial where one term is negative, such as \(8x^2y - 4xy + 2\)?
4. How do you factor an expression when the GCF is a variable only, for example, \(x^3 + x^2 + x\)?
5. What is the next step after factoring using the GCF, if the remaining expression can still be factored?
6. How would you apply the GCF method to an expression with four terms?
7. Can you factor the expression \(12x^3y^2 - 9x^2y + 3xy\) using the GCF?
8. What is the significance of identifying the correct GCF before factoring a polynomial?

**Tip:** When the GCF is just a number (no variables), it simplifies the expression but doesn't reduce the degree of any term.

Learn how to factor the expression 10x²y³ + 20xy + 5 using the Greatest Common Factor (GCF) method with detailed step-by-step instructions. This problem covers polynomial factorization techniques and algebraic manipulations. Suitable for students in Grades 9-12.

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