Divide { [2x^2 + 17x + 21] \div [2x + 3]$}$.Your Answer Should Provide The Quotient And The Remainder.Quotient: { \square$}$Remainder: { \square$}$

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation in mathematics, and understanding how to perform it is essential for solving various mathematical problems. In this article, we will focus on dividing the polynomial $2x^2 + 17x + 21$ by $2x + 3$. Our goal is to find the quotient and the remainder of this division.

The Division Process

To divide the polynomial $2x^2 + 17x + 21$ by $2x + 3$, we will use the long division method. This method involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend.

Step 1: Divide the Highest Degree Term

We start by dividing the highest degree term of the dividend, which is $2x^2$, by the highest degree term of the divisor, which is $2x$. This gives us $x$.

Step 2: Multiply the Divisor by the Result

Next, we multiply the entire divisor, $2x + 3$, by the result we obtained in the previous step, which is $x$. This gives us $2x^2 + 3x$.

Step 3: Subtract the Result from the Dividend

We then subtract the result we obtained in the previous step from the dividend. This gives us $2x^2 + 17x + 21 - (2x^2 + 3x) = 14x + 21$.

Step 4: Repeat the Process

We repeat the process by dividing the highest degree term of the new dividend, which is $14x$, by the highest degree term of the divisor, which is $2x$. This gives us $7$.

Step 5: Multiply the Divisor by the Result

Next, we multiply the entire divisor, $2x + 3$, by the result we obtained in the previous step, which is $7$. This gives us $14x + 21$.

Step 6: Subtract the Result from the Dividend

We then subtract the result we obtained in the previous step from the dividend. This gives us $14x + 21 - (14x + 21) = 0$.

The Quotient and Remainder

Since the final result of the subtraction is $0$, we can conclude that the division is exact, and there is no remainder. The quotient is therefore $x + 7$.

Conclusion

In this article, we have demonstrated how to divide the polynomial $2x^2 + 17x + 21$ by $2x + 3$ using the long division method. We have obtained the quotient, which is $x + 7$, and have shown that there is no remainder. This process is essential for solving various mathematical problems, and understanding how to perform it is crucial for success in mathematics.

Example Problems

Problem 1

Divide the polynomial $3x^2 + 11x + 15$ by $3x + 2$.

Solution

To divide the polynomial $3x^2 + 11x + 15$ by $3x + 2$, we will use the long division method. We start by dividing the highest degree term of the dividend, which is $3x^2$, by the highest degree term of the divisor, which is $3x$. This gives us $x$. We then multiply the entire divisor, $3x + 2$, by the result we obtained in the previous step, which is $x$. This gives us $3x^2 + 2x$. We then subtract the result we obtained in the previous step from the dividend. This gives us $3x^2 + 11x + 15 - (3x^2 + 2x) = 9x + 15$. We repeat the process by dividing the highest degree term of the new dividend, which is $9x$, by the highest degree term of the divisor, which is $3x$. This gives us $3$. We then multiply the entire divisor, $3x + 2$, by the result we obtained in the previous step, which is $3$. This gives us $9x + 6$. We then subtract the result we obtained in the previous step from the dividend. This gives us $9x + 15 - (9x + 6) = 9$. Since the final result of the subtraction is $9$, we can conclude that the division is not exact, and there is a remainder. The quotient is therefore $x + 3$, and the remainder is $9$.

Problem 2

Divide the polynomial $2x^2 + 5x + 6$ by $2x + 1$.

Solution

To divide the polynomial $2x^2 + 5x + 6$ by $2x + 1$, we will use the long division method. We start by dividing the highest degree term of the dividend, which is $2x^2$, by the highest degree term of the divisor, which is $2x$. This gives us $x$. We then multiply the entire divisor, $2x + 1$, by the result we obtained in the previous step, which is $x$. This gives us $2x^2 + x$. We then subtract the result we obtained in the previous step from the dividend. This gives us $2x^2 + 5x + 6 - (2x^2 + x) = 4x + 6$. We repeat the process by dividing the highest degree term of the new dividend, which is $4x$, by the highest degree term of the divisor, which is $2x$. This gives us $2$. We then multiply the entire divisor, $2x + 1$, by the result we obtained in the previous step, which is $2$. This gives us $4x + 2$. We then subtract the result we obtained in the previous step from the dividend. This gives us $4x + 6 - (4x + 2) = 4$. Since the final result of the subtraction is $4$, we can conclude that the division is not exact, and there is a remainder. The quotient is therefore $x + 2$, and the remainder is $4$.

Final Answer

The final answer is:

$$ $$

Introduction

In our previous article, we discussed how to divide polynomials using the long division method. We also provided example problems to help illustrate the process. In this article, we will answer some frequently asked questions about dividing polynomials.

Q&A

Q: What is the purpose of dividing polynomials?

A: The purpose of dividing polynomials is to simplify complex expressions and to find the quotient and remainder of a division operation.

Q: What is the long division method?

A: The long division method is a step-by-step process for dividing polynomials. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend.

Q: How do I know when to stop dividing?

A: You know when to stop dividing when the final result of the subtraction is zero. This indicates that the division is exact, and there is no remainder.

Q: What is the quotient and remainder?

A: The quotient is the result of the division operation, and the remainder is the amount left over after the division operation.

Q: Can I use the long division method to divide any polynomial?

A: Yes, you can use the long division method to divide any polynomial. However, you must make sure that the degree of the divisor is less than or equal to the degree of the dividend.

Q: What if the division is not exact?

A: If the division is not exact, you will have a remainder. The quotient will be the result of the division operation, and the remainder will be the amount left over after the division operation.

Q: Can I use the long division method to divide polynomials with negative coefficients?

A: Yes, you can use the long division method to divide polynomials with negative coefficients. However, you must make sure to follow the same steps as you would with positive coefficients.

Q: Can I use the long division method to divide polynomials with fractional coefficients?

A: Yes, you can use the long division method to divide polynomials with fractional coefficients. However, you must make sure to follow the same steps as you would with integer coefficients.

Q: What if I make a mistake during the long division process?

A: If you make a mistake during the long division process, you can try to correct it by re-evaluating the previous steps. If you are still unsure, you can try to re-do the entire process.

Example Problems

Problem 1

Divide the polynomial $3x^2 + 11x + 15$ by $3x + 2$.

Solution

To divide the polynomial $3x^2 + 11x + 15$ by $3x + 2$, we will use the long division method. We start by dividing the highest degree term of the dividend, which is $3x^2$, by the highest degree term of the divisor, which is $3x$. This gives us $x$. We then multiply the entire divisor, $3x + 2$, by the result we obtained in the previous step, which is $x$. This gives us $3x^2 + 2x$. We then subtract the result we obtained in the previous step from the dividend. This gives us $3x^2 + 11x + 15 - (3x^2 + 2x) = 9x + 15$. We repeat the process by dividing the highest degree term of the new dividend, which is $9x$, by the highest degree term of the divisor, which is $3x$. This gives us $3$. We then multiply the entire divisor, $3x + 2$, by the result we obtained in the previous step, which is $3$. This gives us $9x + 6$. We then subtract the result we obtained in the previous step from the dividend. This gives us $9x + 15 - (9x + 6) = 9$. Since the final result of the subtraction is $9$, we can conclude that the division is not exact, and there is a remainder. The quotient is therefore $x + 3$, and the remainder is $9$.

Problem 2

Divide the polynomial $2x^2 + 5x + 6$ by $2x + 1$.

Solution

To divide the polynomial $2x^2 + 5x + 6$ by $2x + 1$, we will use the long division method. We start by dividing the highest degree term of the dividend, which is $2x^2$, by the highest degree term of the divisor, which is $2x$. This gives us $x$. We then multiply the entire divisor, $2x + 1$, by the result we obtained in the previous step, which is $x$. This gives us $2x^2 + x$. We then subtract the result we obtained in the previous step from the dividend. This gives us $2x^2 + 5x + 6 - (2x^2 + x) = 4x + 6$. We repeat the process by dividing the highest degree term of the new dividend, which is $4x$, by the highest degree term of the divisor, which is $2x$. This gives us $2$. We then multiply the entire divisor, $2x + 1$, by the result we obtained in the previous step, which is $2$. This gives us $4x + 2$. We then subtract the result we obtained in the previous step from the dividend. This gives us $4x + 6 - (4x + 2) = 4$. Since the final result of the subtraction is $4$, we can conclude that the division is not exact, and there is a remainder. The quotient is therefore $x + 2$, and the remainder is $4$.

Final Answer

The final answer is:

Quotient: $\boxed{x + 7}$ Remainder: $\boxed{0}$