The Variables $A, B$, And $C$[/tex\] Represent Polynomials Where $A = X + 1$, $B = X^2 + 2x - 1$, And \$C = 2x$[/tex\]. What Is $AB + C$ In Simplest Form?A. $x^3 + 3x - 1$ B.

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Introduction

In algebra, polynomials are a fundamental concept used to represent various mathematical expressions. Given the variables A, B, and C, each representing a polynomial, we are tasked with finding the simplified form of the expression AB + C. In this article, we will delve into the world of polynomials, explore the given expressions, and simplify the resulting polynomial.

Understanding the Polynomials A, B, and C

To begin, let's examine the given polynomials:

  • A = x + 1
  • B = x^2 + 2x - 1
  • C = 2x

These polynomials are represented in their simplest form, with A being a linear polynomial, B being a quadratic polynomial, and C being a linear polynomial as well.

Multiplying Polynomials A and B

To find the expression AB, we need to multiply the two polynomials A and B. When multiplying polynomials, we follow the distributive property, which states that for any polynomials P(x) and Q(x), the product P(x)Q(x) is obtained by multiplying each term of P(x) by each term of Q(x) and then combining like terms.

Let's multiply A and B:

AB = (x + 1)(x^2 + 2x - 1)

Using the distributive property, we get:

AB = x(x^2 + 2x - 1) + 1(x^2 + 2x - 1)

Expanding the terms, we get:

AB = x^3 + 2x^2 - x + x^2 + 2x - 1

Combining like terms, we get:

AB = x^3 + 3x^2 + x - 1

Adding Polynomial C to AB

Now that we have the expression AB, we can add polynomial C to it:

AB + C = (x^3 + 3x^2 + x - 1) + 2x

Combining like terms, we get:

AB + C = x^3 + 3x^2 + 3x - 1

Conclusion

In conclusion, the simplified form of the expression AB + C is x^3 + 3x^2 + 3x - 1. This polynomial is the result of multiplying polynomials A and B and then adding polynomial C to the resulting expression.

Final Answer

The final answer is x^3 + 3x^2 + 3x - 1.

Discussion

The expression AB + C can be simplified by first multiplying polynomials A and B, and then adding polynomial C to the resulting expression. The distributive property is used to multiply the polynomials, and like terms are combined to simplify the expression. The final answer is x^3 + 3x^2 + 3x - 1.

Related Topics

  • Polynomials
  • Algebra
  • Distributive Property
  • Like Terms

References

  • [1] Algebra, 2nd Edition, Michael Artin
  • [2] Polynomials, 1st Edition, David C. Lay

Note: The references provided are for informational purposes only and are not directly related to the problem at hand.

Introduction

In our previous article, we explored the simplified form of the expression AB + C, where A = x + 1, B = x^2 + 2x - 1, and C = 2x. We found that the simplified form of the expression is x^3 + 3x^2 + 3x - 1. In this article, we will address some of the frequently asked questions related to the topic.

Q&A

Q: What is the distributive property in algebra?

A: The distributive property is a fundamental concept in algebra that states that for any polynomials P(x) and Q(x), the product P(x)Q(x) is obtained by multiplying each term of P(x) by each term of Q(x) and then combining like terms.

Q: How do you multiply polynomials?

A: To multiply polynomials, you follow the distributive property, which involves multiplying each term of one polynomial by each term of the other polynomial and then combining like terms.

Q: What are like terms in algebra?

A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 3x are like terms because they both have the variable x raised to the power of 1.

Q: How do you simplify a polynomial expression?

A: To simplify a polynomial expression, you combine like terms and eliminate any unnecessary terms.

Q: What is the difference between a linear polynomial and a quadratic polynomial?

A: A linear polynomial is a polynomial of degree 1, which means it has only one term with a variable raised to the power of 1. A quadratic polynomial, on the other hand, is a polynomial of degree 2, which means it has two terms with variables raised to the power of 2.

Q: Can you provide an example of a polynomial expression that is not in simplest form?

A: Yes, consider the polynomial expression 2x^2 + 3x + 4x. This expression is not in simplest form because the terms 3x and 4x are like terms and can be combined to form the term 7x.

Q: How do you add polynomials?

A: To add polynomials, you combine like terms and eliminate any unnecessary terms.

Q: What is the final answer to the expression AB + C?

A: The final answer to the expression AB + C is x^3 + 3x^2 + 3x - 1.

Conclusion

In conclusion, the expression AB + C can be simplified by first multiplying polynomials A and B, and then adding polynomial C to the resulting expression. The distributive property is used to multiply the polynomials, and like terms are combined to simplify the expression. We hope this Q&A article has provided you with a better understanding of the topic.

Final Answer

The final answer is x^3 + 3x^2 + 3x - 1.

Discussion

The expression AB + C can be simplified by first multiplying polynomials A and B, and then adding polynomial C to the resulting expression. The distributive property is used to multiply the polynomials, and like terms are combined to simplify the expression. The final answer is x^3 + 3x^2 + 3x - 1.

Related Topics

  • Polynomials
  • Algebra
  • Distributive Property
  • Like Terms

References

  • [1] Algebra, 2nd Edition, Michael Artin
  • [2] Polynomials, 1st Edition, David C. Lay

Note: The references provided are for informational purposes only and are not directly related to the problem at hand.