What Is The Sum Of The Polynomials?${ \left(-x^2 + 9\right) + \left(-3x^2 - 11x + 4\right) }$A. { -4x^2 - 2x + 4$}$B. { -4x^2 - 11x + 13$}$C. ${ 2x^2 + 20x + 4\$} D. ${ 2x^2 + 11x + 5\$}

Understanding Polynomials and Their Operations

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When dealing with polynomials, it's essential to understand the rules of operations, including the addition and subtraction of like terms. In this article, we will explore the concept of adding polynomials and provide a step-by-step guide on how to find the sum of two given polynomials.

The Basics of Adding Polynomials

Adding polynomials involves combining like terms, which are terms that have the same variable and exponent. When adding polynomials, we need to combine the coefficients of like terms. For example, if we have two polynomials: $x^2 + 3x + 4$ and $2x^2 - 5x - 3$, we can add them by combining the like terms.

Step-by-Step Guide to Adding Polynomials

To add the given polynomials, $\left(-x^2 + 9\right) + \left(-3x^2 - 11x + 4\right)$, we need to follow the order of operations and combine like terms.

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to the terms inside the second polynomial. This will change the signs of the terms.

$\left(-x^2 + 9\right) + \left(-3x^2 - 11x + 4\right)$ $= \left(-x^2 + 9\right) + \left(-3x^2\right) + \left(-11x\right) + \left(4\right)$

Step 2: Combine Like Terms

Now, we can combine the like terms. The like terms are the terms that have the same variable and exponent.

$= \left(-x^2 - 3x^2\right) + \left(-11x\right) + \left(4\right) + 9$ $= \left(-4x^2\right) + \left(-11x\right) + \left(13\right)$

Step 3: Simplify the Expression

The final step is to simplify the expression by combining the constants.

$= -4x^2 - 11x + 13$

Conclusion

In conclusion, the sum of the given polynomials, $\left(-x^2 + 9\right) + \left(-3x^2 - 11x + 4\right)$, is $-4x^2 - 11x + 13$. This is the correct answer among the given options.

Final Answer

The final answer is $\boxed{-4x^2 - 11x + 13}$.

Discussion

This problem requires a basic understanding of polynomials and their operations. The key concept is to combine like terms and follow the order of operations. If you have any questions or need further clarification, please feel free to ask.

Related Topics

• Adding and Subtracting Polynomials
• Multiplying and Dividing Polynomials
• Factoring Polynomials
• Quadratic Equations

References

• Polynomial Operations
• Adding and Subtracting Polynomials
• Multiplying and Dividing Polynomials
• Factoring Polynomials
  

Understanding Polynomials and Their Operations

Adding polynomials is a fundamental concept in algebra that involves combining like terms to simplify expressions. In this article, we will address some of the most frequently asked questions on adding polynomials.

Q: What are like terms in polynomials?

A: Like terms are terms that have the same variable and exponent. For example, in the polynomial $x^2 + 3x + 4$, the terms $x^2$ and $3x$ are like terms because they both have the variable $x$ and the exponent $2$.

Q: How do I add polynomials with different variables?

A: When adding polynomials with different variables, we need to combine the like terms separately. For example, if we have the polynomials $x^2 + 3y$ and $2x^2 + 4y$, we can add them by combining the like terms: $x^2 + 2x^2 + 3y + 4y = 3x^2 + 7y$.

Q: Can I add polynomials with negative coefficients?

A: Yes, you can add polynomials with negative coefficients. When adding polynomials with negative coefficients, we need to distribute the negative sign to the terms inside the second polynomial. For example, if we have the polynomials $x^2 + 3x - 4$ and $-2x^2 - 5x + 2$, we can add them by distributing the negative sign: $x^2 + 3x - 4 + (-2x^2 - 5x + 2) = -x^2 - 2x - 2$.

Q: How do I add polynomials with fractions?

A: When adding polynomials with fractions, we need to find a common denominator and then add the numerators. For example, if we have the polynomials $\frac{x^2}{2} + \frac{3x}{4}$ and $\frac{2x^2}{3} + \frac{5x}{6}$, we can add them by finding a common denominator: $\frac{6x^2}{12} + \frac{9x}{12} + \frac{8x^2}{12} + \frac{10x}{12} = \frac{14x^2}{12} + \frac{19x}{12}$.

Q: Can I add polynomials with exponents?

A: Yes, you can add polynomials with exponents. When adding polynomials with exponents, we need to combine the like terms separately. For example, if we have the polynomials $x^2 + 3x^3$ and $2x^2 + 4x^3$, we can add them by combining the like terms: $x^2 + 2x^2 + 3x^3 + 4x^3 = 3x^2 + 7x^3$.

Q: How do I add polynomials with variables and constants?

A: When adding polynomials with variables and constants, we need to combine the like terms separately. For example, if we have the polynomials $x^2 + 3x + 4$ and $2x^2 + 5x + 2$, we can add them by combining the like terms: $x^2 + 2x^2 + 3x + 5x + 4 + 2 = 3x^2 + 8x + 6$.

Q: Can I add polynomials with imaginary numbers?

A: Yes, you can add polynomials with imaginary numbers. When adding polynomials with imaginary numbers, we need to combine the like terms separately. For example, if we have the polynomials $x^2 + 3ix + 4$ and $2x^2 - 5ix + 2$, we can add them by combining the like terms: $x^2 + 2x^2 + 3ix - 5ix + 4 + 2 = 3x^2 - 2ix + 6$.

Conclusion

In conclusion, adding polynomials is a fundamental concept in algebra that involves combining like terms to simplify expressions. By understanding the rules of operations and following the order of operations, we can add polynomials with different variables, negative coefficients, fractions, exponents, variables and constants, and imaginary numbers.

Final Answer

The final answer is $\boxed{3x^2 - 2ix + 6}$.

Discussion

This problem requires a basic understanding of polynomials and their operations. The key concept is to combine like terms and follow the order of operations. If you have any questions or need further clarification, please feel free to ask.

Related Topics

• Adding and Subtracting Polynomials
• Multiplying and Dividing Polynomials
• Factoring Polynomials
• Quadratic Equations

References

• Polynomial Operations
• Adding and Subtracting Polynomials
• Multiplying and Dividing Polynomials
• Factoring Polynomials