Select The Correct Answer.Which Expression Is Equivalent To The Given Polynomial Expression? \left(-4a^2 - 3b\right) + \left(-2ab - A^2 + B^2\right) + \left(-b^2 + 6ab\right ]A. − 3 A 2 + 2 B 2 + 8 A B + 3 B -3a^2 + 2b^2 + 8ab + 3b − 3 A 2 + 2 B 2 + 8 Ab + 3 B B. $-5a^2 + 2b^2 + 8ab +
**Simplifying Polynomial Expressions: A Step-by-Step Guide** ===========================================================
Introduction
Polynomial expressions are a fundamental concept in algebra, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying polynomial expressions, focusing on combining like terms and rearranging the expression to its simplest form.
What are Polynomial Expressions?
A polynomial expression is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It can be written in the form:
a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0
where a_n, a_(n-1), ..., a_1, a_0 are coefficients, and x is the variable.
Simplifying Polynomial Expressions: A Step-by-Step Guide
Step 1: Combine Like Terms
Like terms are terms that have the same variable raised to the same power. To combine like terms, we add or subtract the coefficients of the like terms.
Example:
Simplify the expression: 2x^2 + 3x^2 + 4x
Combine like terms: (2 + 3)x^2 + 4x = 5x^2 + 4x
Step 2: Rearrange the Expression
Once we have combined like terms, we can rearrange the expression to put the terms in descending order of the variable's exponent.
Example:
Simplify the expression: 5x^2 + 4x
Rearrange the expression: 5x^2 + 4x = 5x^2 + 4x^1
Step 3: Simplify the Expression
Now that we have combined like terms and rearranged the expression, we can simplify it further by combining any remaining like terms.
Example:
Simplify the expression: 5x^2 + 4x^1
Combine like terms: 5x^2 + 4x^1 = 5x^2 + 4x
Simplifying the Given Polynomial Expression
Now, let's apply the steps we learned to simplify the given polynomial expression:
Step 1: Combine Like Terms
Combine like terms: (-4a^2 - a^2) + (-3b + b^2 + b^2) + (-2ab + 6ab)
Simplify: -5a^2 + (-3b + 2b^2) + 4ab
Step 2: Rearrange the Expression
Rearrange the expression: -5a^2 + 2b^2 + 4ab - 3b
Step 3: Simplify the Expression
Combine like terms: -5a^2 + 2b^2 + 4ab - 3b
Conclusion
Simplifying polynomial expressions is a crucial skill in algebra, and by following the steps outlined in this article, you can master the process. Remember to combine like terms, rearrange the expression, and simplify further to put the terms in descending order of the variable's exponent.
Q&A
Q: What are like terms? A: Like terms are terms that have the same variable raised to the same power.
Q: How do I combine like terms? A: To combine like terms, add or subtract the coefficients of the like terms.
Q: What is the order of operations for simplifying polynomial expressions? A: The order of operations is:
- Combine like terms
- Rearrange the expression
- Simplify further
Q: Can I simplify a polynomial expression by combining all the terms at once? A: No, it's better to combine like terms step by step to avoid errors.
Q: How do I know if a term is a like term? A: A term is a like term if it has the same variable raised to the same power.
Q: Can I simplify a polynomial expression with variables and constants? A: Yes, you can simplify a polynomial expression with variables and constants by combining like terms and rearranging the expression.
Q: What is the final answer to the given polynomial expression? A: The final answer is -5a^2 + 2b^2 + 4ab - 3b.