Write the Polynomials in a Vertical Format

Oct 13, 2025 by talynt.org 88 views

Introduction

In algebra, the vertical multiplication method is a technique used to multiply two polynomials. This method is particularly useful when dealing with polynomials of high degree. In this article, we will explore the vertical multiplication method and apply it to a given problem.

What is the Vertical Multiplication Method?

The vertical multiplication method is a step-by-step process used to multiply two polynomials. It involves writing the polynomials in a vertical format, with the terms of each polynomial aligned. The method involves multiplying each term of the first polynomial by each term of the second polynomial and then combining like terms.

Step-by-Step Guide to the Vertical Multiplication Method

Write the Polynomials in a Vertical Format: Write the two polynomials in a vertical format, with the terms of each polynomial aligned.
Multiply Each Term of the First Polynomial by Each Term of the Second Polynomial: Multiply each term of the first polynomial by each term of the second polynomial.
Combine Like Terms: Combine like terms to simplify the result.

Example Problem

Let's apply the vertical multiplication method to the following problem:

{ \begin{array}{r} x^2 + 5x + 1 \\ x \, (3x^2 - 2x + 4) \\ \hline \end{array} \}

Step 1: Write the Polynomials in a Vertical Format

{ \begin{array}{r} x^2 + 5x + 1 \\ x \, (3x^2 - 2x + 4) \\ \hline \end{array} \}

Step 2: Multiply Each Term of the First Polynomial by Each Term of the Second Polynomial

To multiply each term of the first polynomial by each term of the second polynomial, we need to multiply each term of the first polynomial by each term of the second polynomial.

{ \begin{array}{r} x^2 \, (3x^2) & + & x^2 \, (-2x) & + & x^2 \, (4) \\ 5x \, (3x^2) & + & 5x \, (-2x) & + & 5x \, (4) \\ 1 \, (3x^2) & + & 1 \, (-2x) & + & 1 \, (4) \\ \hline \end{array} \}

Step 3: Combine Like Terms

To combine like terms, we need to group together the terms that have the same variable and exponent.

{ \begin{array}{r} 3x^4 & - & 2x^3 & + & 4x^2 \\ 15x^3 & - & 10x^2 & + & 20x \\ 3x^2 & - & 2x & + & 4 \\ \hline \end{array} \}

Simplifying the Result

To simplify the result, we need to combine like terms.

{ \begin{array}{r} 3x^4 & - & 2x^3 & + & 4x^2 \\ 15x^3 & - & 10x^2 & + & 20x \\ 3x^2 & - & 2x & + & 4 \\ \hline \end{array} \}

{ \begin{array}{r} 3x^4 & + & 13x^3 & + & 3x^2 & + & 18x & + & 4 \\ \hline \end{array} \}

Conclusion

In this article, we have explored the vertical multiplication method and applied it to a given problem. The vertical multiplication method is a step-by-step process used to multiply two polynomials. It involves writing the polynomials in a vertical format, multiplying each term of the first polynomial by each term of the second polynomial, and combining like terms. By following these steps, we can simplify the result and obtain the final answer.

Answer

The final answer is:

{ 3x^4 + 13x^3 + 3x^2 + 18x + 4 }

This answer matches option B.

Discussion

The vertical multiplication method is a useful technique for multiplying two polynomials. It involves writing the polynomials in a vertical format, multiplying each term of the first polynomial by each term of the second polynomial, and combining like terms. By following these steps, we can simplify the result and obtain the final answer.

The vertical multiplication method is particularly useful when dealing with polynomials of high degree. It allows us to multiply two polynomials in a step-by-step process, making it easier to simplify the result and obtain the final answer.

In conclusion, the vertical multiplication method is a useful technique for multiplying two polynomials. It involves writing the polynomials in a vertical format, multiplying each term of the first polynomial by each term of the second polynomial, and combining like terms. By following these steps, we can simplify the result and obtain the final answer.