Factor $20a^3 - 25a^2b + 8ab^2 - 10b^3$.A. $\left(5a^2 + 2b^2\right)(4a - 5b)$B. \$\left(5a^2 - 2b^2\right)(4a + 5b)$[/tex\]C. $\left(5a^2 + 2b\right)(4a - 5b)$D. $\left(20a^2 + 2b^2\right)(a - 5b)$

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Introduction

Factoring a cubic expression can be a challenging task, but with the right approach, it can be broken down into manageable steps. In this article, we will explore the process of factoring a given cubic expression and provide a step-by-step guide on how to do it.

The Given Expression

The given expression is:

20a3−25a2b+8ab2−10b320a^3 - 25a^2b + 8ab^2 - 10b^3

Our goal is to factor this expression into a product of two binomials.

Step 1: Look for Common Factors

The first step in factoring a cubic expression is to look for common factors. In this case, we can factor out a negative sign and a common factor of 5a25a^2 from the first two terms:

20a3−25a2b=5a2(4a−5b)20a^3 - 25a^2b = 5a^2(4a - 5b)

Step 2: Look for Patterns

Next, we need to look for patterns in the remaining terms. We can see that the last two terms have a common factor of 2b22b^2:

8ab2−10b3=2b2(4a−5b)8ab^2 - 10b^3 = 2b^2(4a - 5b)

Step 3: Combine the Factors

Now that we have factored out common factors and identified patterns, we can combine the factors to get the final result:

20a3−25a2b+8ab2−10b3=5a2(4a−5b)+2b2(4a−5b)20a^3 - 25a^2b + 8ab^2 - 10b^3 = 5a^2(4a - 5b) + 2b^2(4a - 5b)

We can now factor out the common binomial (4a−5b)(4a - 5b):

20a3−25a2b+8ab2−10b3=(5a2+2b2)(4a−5b)20a^3 - 25a^2b + 8ab^2 - 10b^3 = (5a^2 + 2b^2)(4a - 5b)

Conclusion

In this article, we have factored the given cubic expression into a product of two binomials. We started by looking for common factors and identifying patterns in the remaining terms. By combining the factors, we were able to get the final result. The correct answer is:

(5a2+2b2)(4a−5b)\left(5a^2 + 2b^2\right)(4a - 5b)

Comparison with Other Options

Let's compare our result with the other options:

  • Option A: $\left(5a^2 + 2b^2\right)(4a - 5b)$
  • Option B: $\left(5a^2 - 2b^2\right)(4a + 5b)$
  • Option C: $\left(5a^2 + 2b\right)(4a - 5b)$
  • Option D: $\left(20a^2 + 2b^2\right)(a - 5b)$

Our result matches option A, which is the correct answer.

Tips and Tricks

Here are some tips and tricks to help you factor cubic expressions:

  • Look for common factors and patterns in the terms.
  • Use the distributive property to expand the expression and identify common factors.
  • Factor out common binomials and combine the factors to get the final result.
  • Check your result by multiplying the factors together to get the original expression.

By following these steps and tips, you should be able to factor cubic expressions with ease.

Real-World Applications

Factoring cubic expressions has many real-world applications, such as:

  • Algebra: Factoring cubic expressions is a fundamental concept in algebra, and it is used to solve equations and inequalities.
  • Calculus: Factoring cubic expressions is used to find the derivative and integral of functions.
  • Physics: Factoring cubic expressions is used to model real-world phenomena, such as the motion of objects.

Introduction

Factoring cubic expressions can be a challenging task, but with the right approach, it can be broken down into manageable steps. In this article, we will provide a Q&A guide to help you understand the process of factoring cubic expressions.

Q: What is a cubic expression?

A: A cubic expression is a polynomial expression of degree 3, which means it has three terms. It is typically written in the form:

ax3+bx2+cx+dax^3 + bx^2 + cx + d

Q: What are the steps to factor a cubic expression?

A: The steps to factor a cubic expression are:

  1. Look for common factors and patterns in the terms.
  2. Use the distributive property to expand the expression and identify common factors.
  3. Factor out common binomials and combine the factors to get the final result.
  4. Check your result by multiplying the factors together to get the original expression.

Q: How do I identify common factors and patterns in the terms?

A: To identify common factors and patterns in the terms, you can look for:

  • Common factors: Look for factors that are common to all the terms, such as a negative sign or a common factor of a2a^2.
  • Patterns: Look for patterns in the terms, such as a common binomial or a pattern of increasing or decreasing coefficients.

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

A: To use the distributive property to expand the expression, you can multiply each term by the other terms. For example, if you have the expression:

2x2+3x+42x^2 + 3x + 4

You can expand it by multiplying each term by the other terms:

2x2(1)+2x2(3x)+2x2(4)+3x(1)+3x(3x)+3x(4)+4(1)+4(3x)+4(4)2x^2(1) + 2x^2(3x) + 2x^2(4) + 3x(1) + 3x(3x) + 3x(4) + 4(1) + 4(3x) + 4(4)

Q: How do I factor out common binomials and combine the factors?

A: To factor out common binomials and combine the factors, you can look for common binomials in the terms and factor them out. For example, if you have the expression:

2x2+3x+42x^2 + 3x + 4

You can factor out the common binomial (2x+1)(2x + 1):

(2x+1)(x+2)(2x + 1)(x + 2)

Q: How do I check my result by multiplying the factors together?

A: To check your result by multiplying the factors together, you can multiply the factors together to get the original expression. For example, if you have the expression:

(2x+1)(x+2)(2x + 1)(x + 2)

You can multiply the factors together to get:

2x2+3x+42x^2 + 3x + 4

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

A: Some common mistakes to avoid when factoring cubic expressions include:

  • Not looking for common factors and patterns in the terms.
  • Not using the distributive property to expand the expression.
  • Not factoring out common binomials and combining the factors.
  • Not checking the result by multiplying the factors together.

Q: How can I practice factoring cubic expressions?

A: You can practice factoring cubic expressions by:

  • Working through examples and exercises in a textbook or online resource.
  • Using online tools and calculators to help you factor expressions.
  • Practicing with real-world problems and applications.

Conclusion

Factoring cubic expressions can be a challenging task, but with the right approach, it can be broken down into manageable steps. By following the steps and tips outlined in this article, you should be able to factor cubic expressions with ease. Remember to look for common factors and patterns in the terms, use the distributive property to expand the expression, factor out common binomials and combine the factors, and check your result by multiplying the factors together.