What Is The Sum Of The Polynomials? \left(7x^3 - 4x^2\right) + \left(2x^3 - 4x^2\right ]A. 5 X 3 5x^3 5 X 3 B. 9 X 3 9x^3 9 X 3 C. 5 X 3 − 8 X 2 5x^3 - 8x^2 5 X 3 − 8 X 2 D. 9 X 3 − 8 X 2 9x^3 - 8x^2 9 X 3 − 8 X 2

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Introduction

In algebra, polynomials are mathematical expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When we are given two or more polynomials, we can combine them by adding or subtracting like terms. In this article, we will explore how to find the sum of two polynomials.

What are Polynomials?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are often represented by letters such as x, y, or z, and the coefficients are numbers that are multiplied with the variables. For example, 2x^2 + 3x - 4 is a polynomial with one variable, x.

Adding Polynomials

When we add two polynomials, we combine like terms. Like terms are terms that have the same variable raised to the same power. For example, 2x^2 and 3x^2 are like terms because they both have the variable x raised to the power of 2.

To add two polynomials, we follow these steps:

  1. Combine like terms by adding or subtracting the coefficients of the like terms.
  2. Write the resulting polynomial in standard form, with the terms arranged in descending order of the powers of the variables.

Example: Adding Two Polynomials

Let's consider the following example:

(7x3−4x2)+(2x3−4x2)\left(7x^3 - 4x^2\right) + \left(2x^3 - 4x^2\right)

To find the sum of these two polynomials, we combine like terms:

  • The like terms are 7x37x^3 and 2x32x^3, so we add their coefficients: 7x3+2x3=9x37x^3 + 2x^3 = 9x^3.
  • The like terms are −4x2-4x^2 and −4x2-4x^2, so we add their coefficients: −4x2+(−4x2)=−8x2-4x^2 + (-4x^2) = -8x^2.

Therefore, the sum of the two polynomials is:

9x3−8x29x^3 - 8x^2

Conclusion

In this article, we explored how to find the sum of two polynomials. We learned that when adding polynomials, we combine like terms by adding or subtracting the coefficients of the like terms. We also saw an example of how to add two polynomials and found the sum to be 9x3−8x29x^3 - 8x^2.

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 we add polynomials?

A: To add polynomials, we combine like terms by adding or subtracting the coefficients of the like terms.

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is a polynomial written with the terms arranged in descending order of the powers of the variables.

Final Answer

The final answer is 9x3−8x2\boxed{9x^3 - 8x^2}.

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Q&A: Polynomials

Q: What is a polynomial?

A polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What are the different types of polynomials?

There are several types of polynomials, including:

  • Monomials: A monomial is a polynomial with only one term. For example, 3x^2 is a monomial.
  • Binomials: A binomial is a polynomial with two terms. For example, 2x^2 + 3x is a binomial.
  • Trinomials: A trinomial is a polynomial with three terms. For example, 2x^2 + 3x + 4 is a trinomial.
  • Polynomials of degree n: A polynomial of degree n is a polynomial with the highest power of the variable equal to n. For example, 2x^3 + 3x^2 + 4x + 5 is a polynomial of degree 3.

Q: How do we add polynomials?

To add polynomials, we combine like terms by adding or subtracting the coefficients of the like terms.

Q: What are like terms in polynomials?

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

Q: How do we subtract polynomials?

To subtract polynomials, we change the signs of the terms in the second polynomial and then add the two polynomials.

Q: What is the standard form of a polynomial?

The standard form of a polynomial is a polynomial written with the terms arranged in descending order of the powers of the variables.

Q: How do we multiply polynomials?

To multiply polynomials, we multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.

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

A polynomial is a specific type of algebraic expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. An algebraic expression, on the other hand, can be any mathematical expression that involves variables and constants.

Q: Can a polynomial have a variable raised to a negative power?

Yes, a polynomial can have a variable raised to a negative power. For example, 2x^(-2) is a polynomial with a variable raised to a negative power.

Q: Can a polynomial have a variable raised to a fractional power?

Yes, a polynomial can have a variable raised to a fractional power. For example, 2x^(1/2) is a polynomial with a variable raised to a fractional power.

Conclusion

In this article, we answered some frequently asked questions about polynomials. We covered topics such as the definition of a polynomial, the different types of polynomials, how to add and subtract polynomials, and the standard form of a polynomial.

Final Answer

The final answer is that polynomials are mathematical expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication.