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

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. They are a fundamental concept in mathematics, and understanding how to add polynomials is crucial for solving various mathematical problems.

Adding Polynomials

When adding polynomials, we combine like terms, which are terms that have the same variable raised to the same power. For example, in the expression $x^2 + 3x + 4$, the terms $x^2$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 2.

Step-by-Step Guide to Adding Polynomials

To add polynomials, follow these steps:

1. Identify like terms: Look for terms in both polynomials that have the same variable raised to the same power.
2. Combine like terms: Add the coefficients of like terms. For example, if we have $2x^2 + 3x^2$, we can combine them to get $5x^2$.
3. Write the final expression: Combine all the like terms and write the final expression.

Example: Adding Polynomials

Let's add the polynomials $\left(-x^2 + 9\right)$ and $\left(-3x^2 - 11x + 4\right)$.

Step 1: Identify Like Terms

The like terms in the two polynomials are:

• $-x^2$ and $-3x^2$
• $-11x$ (no like term in the first polynomial)
• $4$ and $9$ (no like term in the first polynomial)

Step 2: Combine Like Terms

Now, let's combine the like terms:

• $-x^2 + (-3x^2) = -4x^2$
• $-11x$ (no like term in the first polynomial)
• $4 + 9 = 13$

Step 3: Write the Final Expression

The final expression is:

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

Answer

The sum of the polynomials $\left(-x^2 + 9\right)$ and $\left(-3x^2 - 11x + 4\right)$ is:

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

Conclusion

Adding polynomials is a crucial concept in mathematics, and understanding how to do it is essential for solving various mathematical problems. By following the steps outlined in this article, you can add polynomials with ease and confidence.

Key Takeaways

• Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• When adding polynomials, combine like terms by adding their coefficients.
• The final expression is the sum of the polynomials.

Frequently Asked Questions

Q: What are like terms in polynomials?

A: Like terms are terms that have the same variable raised to the same power.

Q: How do I combine like terms?

A: Combine like terms by adding their coefficients.

Q: What is the final expression in adding polynomials?

A: The final expression is the sum of the polynomials.

References

• Polynomial Addition
• Algebraic Expressions
• Mathematics
  Polynomial Addition Q&A ==========================

Frequently Asked Questions

Q: What are polynomials?

A: Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What is the difference between a polynomial and an algebraic expression?

A: An algebraic expression is a general term that refers to any expression involving variables and constants, whereas a polynomial is a specific type of algebraic expression where the variables are raised to non-negative integer powers.

Q: How do I identify like terms in polynomials?

A: Like terms are terms that have the same variable raised to the same power. For example, in the expression $x^2 + 3x + 4$, the terms $x^2$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms in polynomials?

A: Combine like terms by adding their coefficients. For example, if we have $2x^2 + 3x^2$, we can combine them to get $5x^2$.

Q: What is the final expression in adding polynomials?

A: The final expression is the sum of the polynomials.

Q: Can I add polynomials with different degrees?

A: Yes, you can add polynomials with different degrees. For example, you can add a polynomial of degree 2 with a polynomial of degree 3.

Q: How do I add polynomials with different variables?

A: You cannot add polynomials with different variables. For example, you cannot add $x^2 + 3x$ and $y^2 + 4y$ because they have different variables.

Q: Can I subtract polynomials?

A: Yes, you can subtract polynomials. To subtract polynomials, you can add the opposite of the second polynomial to the first polynomial.

Q: How do I multiply polynomials?

A: To multiply polynomials, you can use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.

Q: Can I divide polynomials?

A: Yes, you can divide polynomials. To divide polynomials, you can use long division or synthetic division.

Q: What is the difference between polynomial addition and polynomial multiplication?

A: Polynomial addition involves combining like terms, whereas polynomial multiplication involves multiplying each term in the first polynomial by each term in the second polynomial.

Q: Can I add polynomials with complex numbers?

A: Yes, you can add polynomials with complex numbers. To add polynomials with complex numbers, you can add the real parts and the imaginary parts separately.

Q: How do I add polynomials with fractions?

A: To add polynomials with fractions, you can add the numerators and the denominators separately.

Q: Can I add polynomials with negative exponents?

A: No, you cannot add polynomials with negative exponents. Negative exponents are not allowed in polynomials.

Q: How do I add polynomials with zero?

A: When adding polynomials, zero is treated as a constant and can be added to any polynomial.

Q: Can I add polynomials with variables raised to negative powers?

A: No, you cannot add polynomials with variables raised to negative powers. Negative powers are not allowed in polynomials.

Q: How do I add polynomials with variables raised to fractional powers?

A: No, you cannot add polynomials with variables raised to fractional powers. Fractional powers are not allowed in polynomials.

Conclusion

Adding polynomials is a crucial concept in mathematics, and understanding how to do it is essential for solving various mathematical problems. By following the steps outlined in this article, you can add polynomials with ease and confidence.

Key Takeaways

• Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• When adding polynomials, combine like terms by adding their coefficients.
• The final expression is the sum of the polynomials.

References

• Polynomial Addition
• Algebraic Expressions
• Mathematics