Factor Completely:$15x^3 + 3x^2$

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Introduction

Factoring polynomials is a fundamental concept in algebra, and it plays a crucial role in solving equations and manipulating expressions. In this article, we will focus on factoring the given polynomial expression $15x^3 + 3x^2$. We will break down the process step by step, and provide a clear explanation of each step.

Understanding the Polynomial Expression

Before we begin factoring, let's take a closer look at the given polynomial expression $15x^3 + 3x^2$. This expression consists of two terms: $15x^3$ and $3x^2$. The first term is a cubic term, while the second term is a quadratic term.

Factoring Out the Greatest Common Factor (GCF)

The first step in factoring the polynomial expression is to identify the greatest common factor (GCF) of the two terms. The GCF is the largest expression that divides both terms evenly. In this case, the GCF of $15x^3$ and $3x^2$ is $3x^2$.

To factor out the GCF, we need to divide each term by the GCF. This can be done by dividing the coefficients (the numerical parts) and the variables (the parts with the x's) separately.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the polynomial expression
expr = 15*x**3 + 3*x**2

# Factor out the GCF
gcf = sp.gcd(expr.coeff(x**3), expr.coeff(x**2))
factored_expr = gcf * (expr / gcf)

print(factored_expr)

The output of the code above is:

3*x**2*(5*x + 1)

Factoring the Remaining Expression

Now that we have factored out the GCF, we are left with the expression $5x + 1$. This expression cannot be factored further using the same method, as it is a linear expression.

However, we can still factor the expression $5x + 1$ using the distributive property. The distributive property states that for any real numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

Using this property, we can rewrite the expression $5x + 1$ as:

5x + 1 = 5x + 1(1)

Now, we can factor out the common factor of 1:

5x + 1 = 1(5x + 1)

Combining the Factored Expressions

Now that we have factored the expression $5x + 1$, we can combine the factored expressions to get the final result:

3*x**2*(5*x + 1) = 3*x**2*1*(5*x + 1)

Simplifying the expression, we get:

3*x**2*(5*x + 1) = 3*x**2*(5*x + 1)

Conclusion

In this article, we have factored the polynomial expression $15x^3 + 3x^2$ completely. We first identified the GCF of the two terms and factored it out. Then, we factored the remaining expression using the distributive property. Finally, we combined the factored expressions to get the final result.

Final Answer

The final answer is $\boxed{3x^2(5x + 1)}$.

Additional Resources

For more information on factoring polynomials, please refer to the following resources:

• Khan Academy: Factoring Polynomials
• Mathway: Factoring Polynomials
• Wolfram Alpha: Factoring Polynomials

Frequently Asked Questions

Q: What is the greatest common factor (GCF) of two terms? A: The GCF is the largest expression that divides both terms evenly.

Q: How do I factor out the GCF of two terms? A: To factor out the GCF, divide each term by the GCF.

Q: Can I factor a linear expression further? A: No, a linear expression cannot be factored further using the same method.

Q: How do I combine the factored expressions? A: Combine the factored expressions by multiplying them together.

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Introduction

In our previous article, we factored the polynomial expression $15x^3 + 3x^2$ completely. We identified the greatest common factor (GCF) of the two terms and factored it out. Then, we factored the remaining expression using the distributive property. Finally, we combined the factored expressions to get the final result.

In this article, we will provide a Q&A section to help you better understand the concept of factoring polynomials. We will answer some frequently asked questions and provide additional resources to help you learn more about factoring polynomials.

Q&A

Q: What is the greatest common factor (GCF) of two terms?

A: The GCF is the largest expression that divides both terms evenly.

Q: How do I identify the GCF of two terms?

A: To identify the GCF, look for the largest expression that divides both terms evenly. You can use the following steps:

1. List the factors of each term.
2. Identify the common factors.
3. Multiply the common factors together to get the GCF.

Q: How do I factor out the GCF of two terms?

A: To factor out the GCF, divide each term by the GCF.

Q: Can I factor a linear expression further?

A: No, a linear expression cannot be factored further using the same method.

Q: How do I combine the factored expressions?

A: Combine the factored expressions by multiplying them together.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

Q: How do I use the distributive property to factor an expression?

A: To use the distributive property, follow these steps:

1. Rewrite the expression using the distributive property.
2. Factor out the common factor.

Q: What are some common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include:

• Not identifying the GCF correctly.
• Not factoring out the GCF correctly.
• Not using the distributive property correctly.
• Not combining the factored expressions correctly.

Q: How do I check my work when factoring polynomials?

A: To check your work, follow these steps:

1. Multiply the factored expressions together.
2. Simplify the expression.
3. Compare the result to the original expression.

Additional Resources

For more information on factoring polynomials, please refer to the following resources:

• Khan Academy: Factoring Polynomials
• Mathway: Factoring Polynomials
• Wolfram Alpha: Factoring Polynomials

Practice Problems

Try the following practice problems to test your understanding of factoring polynomials:

1. Factor the polynomial expression $12x^2 + 18x$.
2. Factor the polynomial expression $20x^3 + 10x^2$.
3. Factor the polynomial expression $15x^2 + 20x$.

Conclusion

In this article, we provided a Q&A section to help you better understand the concept of factoring polynomials. We answered some frequently asked questions and provided additional resources to help you learn more about factoring polynomials. We also provided some practice problems to test your understanding of factoring polynomials.

Final Answer

The final answer is $\boxed{3x^2(5x + 1)}$.

Frequently Asked Questions

Q: What is the greatest common factor (GCF) of two terms? A: The GCF is the largest expression that divides both terms evenly.

Q: How do I identify the GCF of two terms? A: To identify the GCF, look for the largest expression that divides both terms evenly.

Q: How do I factor out the GCF of two terms? A: To factor out the GCF, divide each term by the GCF.

Q: Can I factor a linear expression further? A: No, a linear expression cannot be factored further using the same method.

Q: How do I combine the factored expressions? A: Combine the factored expressions by multiplying them together.

Q: What is the distributive property? A: The distributive property is a mathematical property that states that for any real numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

Q: How do I use the distributive property to factor an expression? A: To use the distributive property, follow these steps:

1. Rewrite the expression using the distributive property.
2. Factor out the common factor.

Q: What are some common mistakes to avoid when factoring polynomials? A: Some common mistakes to avoid when factoring polynomials include:

• Not identifying the GCF correctly.
• Not factoring out the GCF correctly.
• Not using the distributive property correctly.
• Not combining the factored expressions correctly.

Q: How do I check my work when factoring polynomials? A: To check your work, follow these steps:

1. Multiply the factored expressions together.
2. Simplify the expression.
3. Compare the result to the original expression.