What Is The Sum Of The Polynomials?$\[ \left(8x^2 - 9y^2 - 4x\right) + \left(x^2 - 3y^2 - 7x\right) \\]A. \[$7x^2 - 6y^2 + 3x\$\]B. \[$9x^2 - 6y^2 + 3x\$\]C. \[$9x^2 - 12y^2 + 3x\$\]D. \[$9x^2 - 12y^2 - 11x\$\]

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Understanding Polynomial Addition

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When adding polynomials, we combine like terms, which are terms that have the same variable raised to the same power. In this article, we will explore how to add two given polynomials and find their sum.

The Given Polynomials

The two polynomials given are:

(8x2−9y2−4x)+(x2−3y2−7x)\left(8x^2 - 9y^2 - 4x\right) + \left(x^2 - 3y^2 - 7x\right)

To find the sum of these polynomials, we need to combine like terms.

Combining Like Terms

Like terms are terms that have the same variable raised to the same power. In the given polynomials, the like terms are:

  • 8x28x^2 and x2x^2
  • −9y2-9y^2 and −3y2-3y^2
  • −4x-4x and −7x-7x

We can combine these like terms by adding their coefficients.

Adding Coefficients

The coefficients of the like terms are:

  • 8x28x^2 and x2x^2: 8+1=98 + 1 = 9
  • −9y2-9y^2 and −3y2-3y^2: −9−3=−12-9 - 3 = -12
  • −4x-4x and −7x-7x: −4−7=−11-4 - 7 = -11

Now, we can rewrite the polynomials with the combined like terms:

(9x2−12y2−11x)\left(9x^2 - 12y^2 - 11x\right)

The Sum of the Polynomials

The sum of the given polynomials is:

(8x2−9y2−4x)+(x2−3y2−7x)=(9x2−12y2−11x)\left(8x^2 - 9y^2 - 4x\right) + \left(x^2 - 3y^2 - 7x\right) = \left(9x^2 - 12y^2 - 11x\right)

Conclusion

In this article, we learned how to add two polynomials by combining like terms. We identified the like terms in the given polynomials and added their coefficients to find the sum. The sum of the polynomials is (9x2−12y2−11x)\left(9x^2 - 12y^2 - 11x\right).

Answer

The correct answer is:

Frequently Asked Questions

In this article, we will answer some frequently asked questions about adding polynomials. Whether you are a student, teacher, or simply someone who wants to learn more about polynomials, this article is for you.

Q: What are like terms in polynomials?

A: Like terms in polynomials are terms that have the same variable raised to the same power. For example, 2x22x^2 and 5x25x^2 are like terms because they both have the variable xx raised to the power of 2.

Q: How do I combine like terms in polynomials?

A: To combine like terms in polynomials, you add the coefficients of the like terms. For example, if you have 2x22x^2 and 5x25x^2, you would add the coefficients 2 and 5 to get 7x27x^2.

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

A: Adding polynomials involves combining like terms, as we discussed earlier. Multiplying polynomials, on the other hand, involves multiplying each term in one polynomial by each term in the other polynomial. For example, if you have the polynomials x2+3xx^2 + 3x and 2x+12x + 1, you would multiply each term in the first polynomial by each term in the second polynomial to get (x2)(2x)+(x2)(1)+(3x)(2x)+(3x)(1)(x^2)(2x) + (x^2)(1) + (3x)(2x) + (3x)(1).

Q: Can I add polynomials with different variables?

A: Yes, you can add polynomials with different variables. For example, if you have the polynomials x2+3y2x^2 + 3y^2 and 2x2+4z22x^2 + 4z^2, you can add them by combining like terms. However, you cannot add polynomials with different variables if they have the same variable raised to the same power. For example, you cannot add x2+3y2x^2 + 3y^2 and 2x2+4x22x^2 + 4x^2 because they have the same variable xx raised to the same power.

Q: How do I simplify a polynomial after adding it to another polynomial?

A: To simplify a polynomial after adding it to another polynomial, you need to combine like terms. This involves adding the coefficients of the like terms and simplifying the resulting expression.

Q: Can I use a calculator to add polynomials?

A: Yes, you can use a calculator to add polynomials. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q: What are some common mistakes to avoid when adding polynomials?

A: Some common mistakes to avoid when adding polynomials include:

  • Forgetting to combine like terms
  • Adding coefficients incorrectly
  • Not simplifying the resulting expression
  • Using a calculator without checking your work by hand

Conclusion

In this article, we answered some frequently asked questions about adding polynomials. We discussed like terms, combining like terms, and simplifying polynomials after adding them to another polynomial. We also covered some common mistakes to avoid when adding polynomials. Whether you are a student, teacher, or simply someone who wants to learn more about polynomials, we hope this article has been helpful.