If $m(x) = X^2 + 3$ And $n(x) = 5x + 9$, Which Expression Is Equivalent To \[$(mn)(x)\$\]?A. $5x^3 + 9x^2 + 15x + 27$B. $25x^2 + 90x + 84$C. $x^2 + 5x + 12$D. $5x^2 + 24$

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Introduction

In algebra, polynomial multiplication is a fundamental concept that involves multiplying two or more polynomials to obtain a new polynomial. In this article, we will explore the concept of polynomial multiplication and apply it to a specific problem involving two given polynomials, m(x)=x2+3m(x) = x^2 + 3 and n(x)=5x+9n(x) = 5x + 9. Our goal is to determine which expression is equivalent to the product of these two polynomials, denoted as (mn)(x)(mn)(x).

What is Polynomial Multiplication?

Polynomial multiplication is a process of multiplying two or more polynomials to obtain a new polynomial. This process involves multiplying each term of one polynomial by each term of the other polynomial and combining like terms. The resulting polynomial is a new expression that represents the product of the original polynomials.

The Problem: Finding the Product of Two Polynomials

Given the two polynomials m(x)=x2+3m(x) = x^2 + 3 and n(x)=5x+9n(x) = 5x + 9, we need to find the product of these two polynomials, denoted as (mn)(x)(mn)(x). To do this, we will multiply each term of m(x)m(x) by each term of n(x)n(x) and combine like terms.

Step 1: Multiply Each Term of m(x)m(x) by Each Term of n(x)n(x)

To find the product of m(x)m(x) and n(x)n(x), we will multiply each term of m(x)m(x) by each term of n(x)n(x). This will result in a new polynomial with multiple terms.

  • Multiply x2x^2 by 5x5x: x2â‹…5x=5x3x^2 \cdot 5x = 5x^3
  • Multiply x2x^2 by 99: x2â‹…9=9x2x^2 \cdot 9 = 9x^2
  • Multiply 33 by 5x5x: 3â‹…5x=15x3 \cdot 5x = 15x
  • Multiply 33 by 99: 3â‹…9=273 \cdot 9 = 27

Step 2: Combine Like Terms

Now that we have multiplied each term of m(x)m(x) by each term of n(x)n(x), we need to combine like terms. This involves adding or subtracting terms with the same variable and exponent.

  • Combine the terms with x3x^3: 5x35x^3
  • Combine the terms with x2x^2: 9x29x^2
  • Combine the terms with xx: 15x15x
  • Combine the constant terms: 2727

The Final Answer: (mn)(x)=5x3+9x2+15x+27(mn)(x) = 5x^3 + 9x^2 + 15x + 27

After multiplying each term of m(x)m(x) by each term of n(x)n(x) and combining like terms, we have found the product of the two polynomials, denoted as (mn)(x)(mn)(x). This expression is equivalent to:

(mn)(x)=5x3+9x2+15x+27(mn)(x) = 5x^3 + 9x^2 + 15x + 27

Conclusion

In this article, we have explored the concept of polynomial multiplication and applied it to a specific problem involving two given polynomials, m(x)=x2+3m(x) = x^2 + 3 and n(x)=5x+9n(x) = 5x + 9. We have found the product of these two polynomials, denoted as (mn)(x)(mn)(x), and determined that the correct expression is 5x3+9x2+15x+275x^3 + 9x^2 + 15x + 27.

Answer Key

Based on our calculations, the correct answer is:

A. 5x3+9x2+15x+275x^3 + 9x^2 + 15x + 27

This expression is equivalent to the product of the two polynomials, (mn)(x)(mn)(x).

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Introduction

In our previous article, we explored the concept of polynomial multiplication and applied it to a specific problem involving two given polynomials, m(x)=x2+3m(x) = x^2 + 3 and n(x)=5x+9n(x) = 5x + 9. In this article, we will provide a Q&A guide to help you better understand polynomial multiplication and how to apply it to different problems.

Q: What is polynomial multiplication?

A: Polynomial multiplication is a process of multiplying two or more polynomials to obtain a new polynomial. This process involves multiplying each term of one polynomial by each term of the other polynomial and combining like terms.

Q: How do I multiply two polynomials?

A: To multiply two polynomials, you need to multiply each term of one polynomial by each term of the other polynomial and combine like terms. This can be done using the distributive property, which states that a(b + c) = ab + ac.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a(b + c) = ab + ac. This property allows us to multiply a single term by two or more terms and combine the results.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract terms with the same variable and exponent. For example, if you have the terms 2x and 3x, you can combine them by adding their coefficients: 2x + 3x = 5x.

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

A: Polynomial multiplication involves multiplying two or more polynomials to obtain a new polynomial, while polynomial addition involves adding two or more polynomials to obtain a new polynomial.

Q: Can I multiply a polynomial by a constant?

A: Yes, you can multiply a polynomial by a constant. This is done by multiplying each term of the polynomial by the constant.

Q: How do I multiply a polynomial by a binomial?

A: To multiply a polynomial by a binomial, you need to multiply each term of the polynomial by each term of the binomial and combine like terms.

Q: What is the FOIL method?

A: The FOIL method is a technique used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, which refers to the order in which you multiply the terms.

Q: How do I use the FOIL method?

A: To use the FOIL method, you need to multiply the first terms of each binomial, then the outer terms, then the inner terms, and finally the last terms. You then combine like terms to obtain the final result.

Q: Can I use the FOIL method to multiply a polynomial by a binomial?

A: No, the FOIL method is used to multiply two binomials, not a polynomial by a binomial.

Conclusion

In this article, we have provided a Q&A guide to help you better understand polynomial multiplication and how to apply it to different problems. We have covered topics such as the distributive property, combining like terms, and the FOIL method.

Answer Key

Based on our Q&A guide, here are the answers to the questions:

  • Q: What is polynomial multiplication? A: Polynomial multiplication is a process of multiplying two or more polynomials to obtain a new polynomial.
  • Q: How do I multiply two polynomials? A: To multiply two polynomials, you need to multiply each term of one polynomial by each term of the other polynomial and combine like terms.
  • Q: What is the distributive property? A: The distributive property is a mathematical property that states that a(b + c) = ab + ac.
  • Q: How do I combine like terms? A: To combine like terms, you need to add or subtract terms with the same variable and exponent.
  • Q: What is the difference between polynomial multiplication and polynomial addition? A: Polynomial multiplication involves multiplying two or more polynomials to obtain a new polynomial, while polynomial addition involves adding two or more polynomials to obtain a new polynomial.
  • Q: Can I multiply a polynomial by a constant? A: Yes, you can multiply a polynomial by a constant.
  • Q: How do I multiply a polynomial by a binomial? A: To multiply a polynomial by a binomial, you need to multiply each term of the polynomial by each term of the binomial and combine like terms.
  • Q: What is the FOIL method? A: The FOIL method is a technique used to multiply two binomials.
  • Q: How do I use the FOIL method? A: To use the FOIL method, you need to multiply the first terms of each binomial, then the outer terms, then the inner terms, and finally the last terms. You then combine like terms to obtain the final result.
  • Q: Can I use the FOIL method to multiply a polynomial by a binomial? A: No, the FOIL method is used to multiply two binomials, not a polynomial by a binomial.