Perform The Operation:$\left(2x^2 - 6x - 5\right) + \left(-9x^2 + 5x - 4\right$\]

Introduction

In algebra, polynomial addition is a fundamental operation that involves combining two or more polynomials by adding their corresponding terms. This operation is essential in solving equations, graphing functions, and simplifying expressions. In this article, we will focus on performing polynomial addition using the given expression: $\left(2x^2 - 6x - 5\right) + \left(-9x^2 + 5x - 4\right)$. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding Polynomials

Before we proceed with the addition, it's essential to understand what polynomials are. A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are typically represented by letters such as x, y, or z, while the coefficients are numbers that multiply the variables.

The Given Expression

The given expression is a sum of two polynomials:

$\left(2x^2 - 6x - 5\right) + \left(-9x^2 + 5x - 4\right)$

To add these polynomials, we need to combine like terms, which are terms that have the same variable and exponent.

Step 1: Identify Like Terms

The first step in adding polynomials is to identify like terms. In the given expression, we have two polynomials with the following terms:

• $2x^2$ and $-9x^2$
• $-6x$ and $5x$
• $-5$ and $-4$

These terms are like terms because they have the same variable (x) and exponent (2, 1, and 0, respectively).

Step 2: Combine Like Terms

Now that we have identified the like terms, we can combine them by adding their coefficients. The coefficients are the numbers that multiply the variables.

• $2x^2 + (-9x^2) = (2 - 9)x^2 = -7x^2$
• $-6x + 5x = (-6 + 5)x = -x$
• $-5 + (-4) = (-5 - 4) = -9$

Step 3: Write the Result

Now that we have combined the like terms, we can write the result of the addition.

$\left(2x^2 - 6x - 5\right) + \left(-9x^2 + 5x - 4\right) = -7x^2 - x - 9$

Conclusion

Performing polynomial addition is a straightforward process that involves identifying like terms and combining them by adding their coefficients. By following the steps outlined in this article, you can add polynomials with ease. Remember to always identify like terms and combine them by adding their coefficients.

Example Problems

Here are a few example problems to help you practice adding polynomials:

1. $\left(3x^2 + 2x - 1\right) + \left(-2x^2 - 3x + 2\right)$
2. $\left(x^2 - 4x + 3\right) + \left(-x^2 + 2x - 1\right)$
3. $\left(2x^2 + 5x - 2\right) + \left(-3x^2 - 2x + 1\right)$

Try solving these problems on your own, and then check your answers with the solutions provided below.

Solutions

1. $\left(3x^2 + 2x - 1\right) + \left(-2x^2 - 3x + 2\right) = (3 - 2)x^2 + (2 - 3)x + (-1 + 2) = x^2 - x + 1$
2. $\left(x^2 - 4x + 3\right) + \left(-x^2 + 2x - 1\right) = (1 - 1)x^2 + (-4 + 2)x + (3 - 1) = -2x + 2$
3. $\left(2x^2 + 5x - 2\right) + \left(-3x^2 - 2x + 1\right) = (2 - 3)x^2 + (5 - 2)x + (-2 + 1) = -x^2 + 3x - 1$

Tips and Tricks

Here are a few tips and tricks to help you add polynomials like a pro:

• Always identify like terms before combining them.
• Use parentheses to group like terms together.
• Combine like terms by adding their coefficients.
• Simplify the expression by combining like terms.

By following these tips and tricks, you can add polynomials with ease and become a master of algebra.

Conclusion

Q: What is the first step in adding polynomials?

A: The first step in adding polynomials is to identify like terms. Like terms are terms that have the same variable and exponent.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable and exponent. For example, in the expression $2x^2 + 3x^2$, the terms $2x^2$ and $3x^2$ are like terms because they have the same variable (x) and exponent (2).

Q: What is the next step after identifying like terms?

A: After identifying like terms, the next step is to combine them by adding their coefficients. The coefficients are the numbers that multiply the variables.

Q: How do I combine like terms?

A: To combine like terms, add their coefficients. For example, in the expression $2x^2 + 3x^2$, the coefficients are 2 and 3. Adding these coefficients gives $2 + 3 = 5$, so the combined term is $5x^2$.

Q: What if I have a negative coefficient?

A: If you have a negative coefficient, simply add it to the other coefficient. For example, in the expression $-2x^2 + 3x^2$, the coefficients are -2 and 3. Adding these coefficients gives $-2 + 3 = 1$, so the combined term is $x^2$.

Q: Can I add polynomials with different variables?

A: No, you cannot add polynomials with different variables. For example, the expression $2x^2 + 3y^2$ cannot be added to the expression $4x^2 + 5y^2$ because they have different variables (x and y).

Q: What is the final step in adding polynomials?

A: The final step in adding polynomials is to write the result. This involves combining all the like terms and simplifying the expression.

Q: How do I simplify the expression?

A: To simplify the expression, combine all the like terms and eliminate any unnecessary terms. For example, in the expression $2x^2 + 3x^2 + 4x^2$, the like terms are $2x^2$, $3x^2$, and $4x^2$. Combining these terms gives $9x^2$, so the simplified expression is $9x^2$.

Q: Can I add polynomials with different exponents?

A: Yes, you can add polynomials with different exponents. For example, the expression $2x^2 + 3x^3$ can be added to the expression $4x^2 + 5x^3$ because they have the same variable (x) but different exponents (2 and 3).

Q: What is the difference between adding polynomials and multiplying polynomials?

A: The main difference between adding polynomials and multiplying polynomials is that when you add polynomials, you are combining like terms, whereas when you multiply polynomials, you are multiplying each term by each other term.

Q: Can I add polynomials with fractions?

A: Yes, you can add polynomials with fractions. For example, the expression $\frac{1}{2}x^2 + \frac{3}{4}x^2$ can be added to the expression $\frac{2}{3}x^2 + \frac{5}{6}x^2$ by finding a common denominator and adding the numerators.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is $\left(2x^2 - 6x - 5\right) + \left(-9x^2 + 5x - 4\right) = -7x^2 - x - 9$.