Choose The Correct Simplification Of \[$(6x-5)(2x^2-3x-6)\$\]:A. \[$12x^3 + 28x^2 + 21x + 30\$\] B. \[$12x^3 - 28x^2 - 21x + 30\$\] C. \[$12x^3 + 28x^2 - 21x + 30\$\] D. \[$12x^3 - 28x^2 - 21x - 30\$\]
Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the expression {(6x-5)(2x^2-3x-6)$}$, and we will explore the different methods and techniques used to simplify algebraic expressions.
Understanding the Expression
The given expression is a product of two binomials, {(6x-5)$}$ and {(2x^2-3x-6)$}$. To simplify this expression, we need to multiply each term in the first binomial by each term in the second binomial.
Multiplying Binomials
When multiplying binomials, we can use the distributive property, which states that for any real numbers a, b, c, and d:
a(b + c) = ab + ac
Using this property, we can multiply each term in the first binomial by each term in the second binomial.
Step 1: Multiply the First Term in the First Binomial by Each Term in the Second Binomial
The first term in the first binomial is 6x. We will multiply this term by each term in the second binomial:
6x(2x^2) = 12x^3 6x(-3x) = -18x^2 6x(-6) = -36x
Step 2: Multiply the Second Term in the First Binomial by Each Term in the Second Binomial
The second term in the first binomial is -5. We will multiply this term by each term in the second binomial:
-5(2x^2) = -10x^2 -5(-3x) = 15x -5(-6) = 30
Step 3: Combine Like Terms
Now that we have multiplied each term in the first binomial by each term in the second binomial, we can combine like terms. Like terms are terms that have the same variable and exponent.
The terms 12x^3 and -10x^2 are like terms, as are the terms -18x^2 and -10x^2. We can combine these terms by adding or subtracting their coefficients.
12x^3 + (-10x^2) = 12x^3 - 10x^2 -18x^2 + (-10x^2) = -28x^2
The terms 15x and -36x are like terms, as are the terms 30 and 0. We can combine these terms by adding or subtracting their coefficients.
15x + (-36x) = -21x 30 + 0 = 30
Simplifying the Expression
Now that we have combined like terms, we can simplify the expression by combining the terms we have obtained.
The simplified expression is:
12x^3 - 28x^2 - 21x + 30
Conclusion
In this article, we have simplified the expression {(6x-5)(2x^2-3x-6)$}$ using the distributive property and combining like terms. We have also explored the different methods and techniques used to simplify algebraic expressions.
Answer
The correct simplification of the expression {(6x-5)(2x^2-3x-6)$}$ is:
12x^3 - 28x^2 - 21x + 30
This is option C.
Discussion
Do you have any questions or comments about simplifying algebraic expressions? Do you have any other questions about mathematics? Please feel free to ask, and I will do my best to help.
References
Related Articles
- Simplifying Rational Expressions
- Simplifying Radical Expressions
- Simplifying Exponential Expressions
Simplifying Algebraic Expressions: A Q&A Guide =====================================================
Introduction
In our previous article, we explored the concept of simplifying algebraic expressions, focusing on the expression {(6x-5)(2x^2-3x-6)$}$. We used the distributive property and combining like terms to simplify the expression. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.
Q&A
Q: What is the distributive property?
A: The distributive property is a mathematical concept that states that for any real numbers a, b, c, and d:
a(b + c) = ab + ac
This property allows us to multiply each term in one binomial by each term in another binomial.
Q: How do I simplify an algebraic expression?
A: To simplify an algebraic expression, you need to follow these steps:
- Multiply each term in one binomial by each term in another binomial using the distributive property.
- Combine like terms by adding or subtracting their coefficients.
- Simplify the resulting expression by combining any remaining like terms.
Q: What are like terms?
A: Like terms are terms that have the same variable and exponent. For example, 2x and 4x are like terms, as are 3x^2 and 5x^2.
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract their coefficients. For example, if you have the terms 2x and 4x, you can combine them by adding their coefficients:
2x + 4x = 6x
Q: What is the difference between a binomial and a polynomial?
A: A binomial is an algebraic expression that consists of two terms, such as 2x + 3 or x^2 - 4. A polynomial is an algebraic expression that consists of three or more terms, such as 2x^2 + 3x - 4 or x^3 + 2x^2 - 3x - 1.
Q: How do I simplify a polynomial expression?
A: To simplify a polynomial expression, you need to follow the same steps as simplifying a binomial expression:
- Multiply each term in one binomial by each term in another binomial using the distributive property.
- Combine like terms by adding or subtracting their coefficients.
- Simplify the resulting expression by combining any remaining like terms.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Failing to distribute the terms correctly
- Failing to combine like terms
- Making errors when adding or subtracting coefficients
- Failing to simplify the resulting expression
Conclusion
In this article, we have answered some frequently asked questions about simplifying algebraic expressions. We have also provided some tips and tricks for simplifying expressions, including the distributive property and combining like terms. By following these steps and avoiding common mistakes, you can simplify algebraic expressions with confidence.
Discussion
Do you have any questions or comments about simplifying algebraic expressions? Do you have any other questions about mathematics? Please feel free to ask, and I will do my best to help.