Simplify The Expression: ${ 4a^3 - A^2b - 36a + 9b }$

Mar 12, 2025 by ADMIN 55 views

**Simplify the Expression: A Comprehensive Guide**

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us to solve complex problems and equations. It involves combining like terms, removing unnecessary components, and rearranging the expression to make it more manageable. In this article, we will focus on simplifying the given expression: $4a^3 - a^2b - 36a + 9b$ . We will break down the expression step by step, using various mathematical techniques to simplify it.

Understanding the Expression

The given expression is a polynomial expression, which consists of several terms with different variables and exponents. The expression is: $4a^3 - a^2b - 36a + 9b$ . To simplify this expression, we need to identify the like terms and combine them.

Like Terms

Like terms are terms that have the same variable and exponent. In the given expression, we can identify the following like terms:

$4a^3$ and $-36a$ have the same variable $a$ but different exponents.
$-a^2b$ and $9b$ have the same variable $b$ but different exponents.

Simplifying the Expression

To simplify the expression, we will combine the like terms. We will start by combining the terms with the same variable and exponent.

Combining Terms with the Same Variable and Exponent

We can combine the terms $4a^3$ and $-36a$ by factoring out the common factor $a$ . We get:

4a^3 - 36a = a(4a^2 - 36)

Similarly, we can combine the terms $-a^2b$ and $9b$ by factoring out the common factor $b$ . We get:

-a^2b + 9b = b(-a^2 + 9)

Simplifying the Expression Further

Now that we have combined the like terms, we can simplify the expression further by rearranging the terms. We can rewrite the expression as:

a(4a^2 - 36) + b(-a^2 + 9)

Factoring Out Common Factors

We can factor out the common factor $4$ from the first term and the common factor $-1$ from the second term. We get:

4a(a^2 - 9) - b(a^2 - 9)

Simplifying the Expression Using the Difference of Squares Formula

We can simplify the expression further by using the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$ . We can rewrite the expression as:

4a(a + 3)(a - 3) - b(a + 3)(a - 3)

Simplifying the Expression Further

Now that we have simplified the expression using the difference of squares formula, we can simplify it further by combining the like terms. We get:

(4a - b)(a + 3)(a - 3)

Conclusion

In this article, we simplified the given expression: $4a^3 - a^2b - 36a + 9b$ . We broke down the expression step by step, using various mathematical techniques to simplify it. We identified the like terms, combined them, and simplified the expression further using the difference of squares formula. The final simplified expression is: $(4a - b)(a + 3)(a - 3)$ . This expression is more manageable and easier to work with.

Final Answer

The final answer is: $(4a - b)(a + 3)(a - 3)$ .

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Glossary

Like terms: Terms that have the same variable and exponent.
Difference of squares formula: A formula that states: $a^2 - b^2 = (a + b)(a - b)$ .
Simplifying an expression: Combining like terms and rearranging the expression to make it more manageable.
Simplify the Expression: A Comprehensive Guide =====================================================

Q&A: Simplifying the Expression

In the previous article, we simplified the given expression: $4a^3 - a^2b - 36a + 9b$ . We broke down the expression step by step, using various mathematical techniques to simplify it. In this article, we will answer some frequently asked questions about simplifying the expression.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. In the given expression, we can identify the following like terms:

$4a^3$ and $-36a$ have the same variable $a$ but different exponents.
$-a^2b$ and $9b$ have the same variable $b$ but different exponents.

Q: How do I combine like terms?

A: To combine like terms, we need to identify the terms with the same variable and exponent. We can then add or subtract the coefficients of these terms. For example, in the given expression, we can combine the terms $4a^3$ and $-36a$ by factoring out the common factor $a$ . We get:

4a^3 - 36a = a(4a^2 - 36)

Q: What is the difference of squares formula?

A: The difference of squares formula is a formula that states: $a^2 - b^2 = (a + b)(a - b)$ . We can use this formula to simplify the expression further. For example, in the given expression, we can rewrite the expression as:

4a(a + 3)(a - 3) - b(a + 3)(a - 3)

Q: How do I simplify the expression using the difference of squares formula?

A: To simplify the expression using the difference of squares formula, we need to identify the terms that can be factored using this formula. We can then factor out the common factor and simplify the expression. For example, in the given expression, we can rewrite the expression as:

(4a - b)(a + 3)(a - 3)

Q: What is the final simplified expression?

A: The final simplified expression is: $(4a - b)(a + 3)(a - 3)$ .

Q: How do I check my work?

A: To check your work, you can plug in some values for the variables and see if the expression simplifies to the expected value. For example, if we plug in $a = 1$ and $b = 2$ , we get:

(4(1) - 2)(1 + 3)(1 - 3) = (2)(4)(-2) = -16

This is the expected value, so we can be confident that our simplified expression is correct.