Given The Polynomial: ${ 4x^3 + 0x^2 - 3x + 4\$} , What Will The First Row Of The Multiplication Be?A. { -20x^3 + 0x^2 + 15x - 20$}$B. ${ 4x^3 + 0x^2 - 6x - 20\$} C. { -20x^3 - 5x^2 + 15x - 20$}$D. [$4x^3 + 0x^2

Mar 7, 2025 by ADMIN 212 views

**Polynomial Multiplication: Understanding the First Row**

Introduction

Polynomial multiplication is a fundamental concept in algebra, and it plays a crucial role in various mathematical operations. When multiplying polynomials, it's essential to understand the process and the resulting products. In this article, we will focus on the first row of polynomial multiplication and explore the correct answer.

What is Polynomial Multiplication?

Polynomial multiplication is the process of multiplying two or more polynomials to obtain a new polynomial. This operation 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 First Row of Polynomial Multiplication

When multiplying polynomials, the first row of the multiplication is obtained by multiplying the first term of the first polynomial by the first term of the second polynomial. In the given polynomial ${ $4x^3 + 0x^2 - 3x + 4\$}$ , the first term is ${ $4x^3\$}$ .

Step 1: Multiply the First Term of the First Polynomial by the First Term of the Second Polynomial

To find the first row of the multiplication, we need to multiply the first term of the first polynomial, ${ $4x^3\$}$ , by the first term of the second polynomial. However, since the second polynomial is not provided, we will assume that the second polynomial is ${$ ax^3 + bx^2 + cx + d$}$.

The product of the first term of the first polynomial and the first term of the second polynomial is ${ $4x^3 \cdot ax^3 = 4a(x^3)^2 = 4a x^6\$}$ .

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

Next, we need to multiply the first term of the first polynomial, ${ $4x^3\$}$ , by the second term of the second polynomial, ${$ bx^2$}$.

The product of the first term of the first polynomial and the second term of the second polynomial is ${ $4x^3 \cdot bx^2 = 4b(x^3)(x^2) = 4b x^5\$}$ .

Step 3: Multiply the First Term of the First Polynomial by the Third Term of the Second Polynomial

We also need to multiply the first term of the first polynomial, ${ $4x^3\$}$ , by the third term of the second polynomial, ${$ cx$}$.

The product of the first term of the first polynomial and the third term of the second polynomial is ${ $4x^3 \cdot cx = 4c(x^3)(x) = 4c x^4\$}$ .

Step 4: Multiply the First Term of the First Polynomial by the Fourth Term of the Second Polynomial

Finally, we need to multiply the first term of the first polynomial, ${ $4x^3\$}$ , by the fourth term of the second polynomial, ${$ d$}$.

The product of the first term of the first polynomial and the fourth term of the second polynomial is ${ $4x^3 \cdot d = 4d(x^3) = 4dx^3\$}$ .

Combining Like Terms

Now that we have multiplied the first term of the first polynomial by each term of the second polynomial, we need to combine like terms. The resulting expression is:

${ $4a x^6 + 4b x^5 + 4c x^4 + (4d + 4dx^3)\$}$

The Correct Answer

Based on the above calculations, the correct answer is:

${ $4a x^6 + 4b x^5 + 4c x^4 + (4d + 4dx^3)\$}$

However, since the second polynomial is not provided, we cannot determine the values of ${$ a$, [ $b\$, \[$ c$, and [$d$. Therefore, we cannot determine the exact expression for the first row of the multiplication.

Conclusion

In conclusion, the first row of polynomial multiplication involves multiplying the first term of the first polynomial by each term of the second polynomial and combining like terms. The resulting expression is a new polynomial that represents the product of the original polynomials. However, without knowing the second polynomial, we cannot determine the exact expression for the first row of the multiplication.

Answer Key

The correct answer is not among the options provided. However, based on the calculations above, the correct answer would be an expression of the form:

[ $4a x^6 + 4b x^5 + 4c x^4 + (4d + 4dx^3)\$}$

Final Thoughts

Q: What is polynomial multiplication?

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

Q: What is the first row of polynomial multiplication?

A: The first row of polynomial multiplication involves multiplying the first term of the first polynomial by each term of the second polynomial and combining like terms.

Q: How do I multiply polynomials?

A: To multiply polynomials, you need to multiply each term of one polynomial by each term of the other polynomial and combine like terms. The resulting polynomial is a new expression that represents the product of the original polynomials.

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 you provide an example of polynomial multiplication?

A: Let's consider the polynomials ${ $2x^2 + 3x - 1\$}$ and ${$ x^2 - 2x + 1$}$. To multiply these polynomials, we need to multiply each term of the first polynomial by each term of the second polynomial and combine like terms.

${$ (2x^2 + 3x - 1) \cdot (x^2 - 2x + 1)$]

[ $= 2x^2 \cdot x^2 - 2x^2 \cdot 2x + 2x^2 \cdot 1 + 3x \cdot x^2 - 3x \cdot 2x + 3x \cdot 1 - 1 \cdot x^2 + 1 \cdot 2x - 1 \cdot 1\$}$

${$ = 2x^4 - 4x^3 + 2x^2 + 3x^3 - 6x^2 + 3x - x^2 + 2x - 1$}$

${$ = 2x^4 - x^3 - 3x^2 + 5x - 1$}$

Q: What is the importance of polynomial multiplication?

A: Polynomial multiplication is an essential concept in algebra and has numerous applications in various fields, including physics, engineering, and computer science. It is used to solve systems of equations, find the roots of polynomials, and perform other mathematical operations.

Q: Can you provide some tips for multiplying polynomials?

A: Here are some tips for multiplying polynomials:

Multiply each term of one polynomial by each term of the other polynomial.
Combine like terms to simplify the resulting polynomial.
Use the distributive property to multiply each term of one polynomial by each term of the other polynomial.
Use a calculator or computer software to simplify the resulting polynomial.

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

A: Here are some common mistakes to avoid when multiplying polynomials:

Failing to combine like terms.
Failing to use the distributive property.
Making errors when multiplying terms.
Failing to simplify the resulting polynomial.

Q: Can you provide some practice problems for multiplying polynomials?

A: Here are some practice problems for multiplying polynomials:

Multiply the polynomials ${$ x^2 + 2x - 1$}$ and ${$ x^2 - 3x + 2$}$.
Multiply the polynomials ${ $2x^2 + 3x - 1\$}$ and ${$ x^2 + 2x - 1$}$.
Multiply the polynomials ${$ x^3 + 2x^2 - 1$}$ and ${$ x^2 - 3x + 2$}$.

Conclusion

In conclusion, polynomial multiplication is an essential concept in algebra that has numerous applications in various fields. It involves multiplying each term of one polynomial by each term of the other polynomial and combining like terms. By following the tips and avoiding common mistakes, you can simplify the resulting polynomial and solve mathematical problems.