Divide $2x^4 - 4x^3 - 11x^2 + 3x - 6$ By $x + 2$.Look At The Work Shown For The Division Problem:$ \begin{array}{rrrrr} 2 & -4 & -11 & 3 & -6 \\ & -4 & 16 & -10 & 14 \\ \hline 2 & -8 & 5 & -7 & 8 \end{array} $The Remainder

Divide $2x^4 - 4x^3 - 11x^2 + 3x - 6$ by $x + 2$

Understanding the Division Problem

When dividing polynomials, we use a process similar to long division of numbers. The given problem is to divide the polynomial $2x^4 - 4x^3 - 11x^2 + 3x - 6$ by $x + 2$. To solve this problem, we will use the long division method.

The Long Division Method

The long division method involves dividing the highest degree term of the dividend by the highest degree term of the divisor. In this case, the highest degree term of the dividend is $2x^4$ and the highest degree term of the divisor is $x$. We will divide $2x^4$ by $x$ to get $2x^3$.

Setting Up the Division Problem

To set up the division problem, we will write the dividend and the divisor in the following format:

\begin{array}{rrrrr} 2 & -4 & -11 & 3 & -6 \ & -4 & 16 & -10 & 14 \ \hline 2 & -8 & 5 & -7 & 8 \end{array}

Performing the Division

To perform the division, we will multiply the divisor by the quotient and subtract the result from the dividend. We will repeat this process until we have no more terms to divide.

Step 1: Divide $2x^4$ by $x$

We will divide $2x^4$ by $x$ to get $2x^3$. We will write this quotient above the division bar.

Step 2: Multiply $x + 2$ by $2x^3$

We will multiply $x + 2$ by $2x^3$ to get $2x^4 + 4x^3$. We will write this product below the division bar.

Step 3: Subtract $2x^4 + 4x^3$ from $2x^4 - 4x^3 - 11x^2 + 3x - 6$

We will subtract $2x^4 + 4x^3$ from $2x^4 - 4x^3 - 11x^2 + 3x - 6$ to get $-8x^3 - 11x^2 + 3x - 6$. We will write this result below the division bar.

Step 4: Divide $-8x^3$ by $x$

We will divide $-8x^3$ by $x$ to get $-8x^2$. We will write this quotient above the division bar.

Step 5: Multiply $x + 2$ by $-8x^2$

We will multiply $x + 2$ by $-8x^2$ to get $-8x^3 - 16x^2$. We will write this product below the division bar.

Step 6: Subtract $-8x^3 - 16x^2$ from $-8x^3 - 11x^2 + 3x - 6$

We will subtract $-8x^3 - 16x^2$ from $-8x^3 - 11x^2 + 3x - 6$ to get $5x^2 + 3x - 6$. We will write this result below the division bar.

Step 7: Divide $5x^2$ by $x$

We will divide $5x^2$ by $x$ to get $5x$. We will write this quotient above the division bar.

Step 8: Multiply $x + 2$ by $5x$

We will multiply $x + 2$ by $5x$ to get $5x^2 + 10x$. We will write this product below the division bar.

Step 9: Subtract $5x^2 + 10x$ from $5x^2 + 3x - 6$

We will subtract $5x^2 + 10x$ from $5x^2 + 3x - 6$ to get $-7x - 6$. We will write this result below the division bar.

Step 10: Divide $-7x$ by $x$

We will divide $-7x$ by $x$ to get $-7$. We will write this quotient above the division bar.

Step 11: Multiply $x + 2$ by $-7$

We will multiply $x + 2$ by $-7$ to get $-7x - 14$. We will write this product below the division bar.

Step 12: Subtract $-7x - 14$ from $-7x - 6$

We will subtract $-7x - 14$ from $-7x - 6$ to get $8$. We will write this result below the division bar.

The Quotient and Remainder

The quotient of the division is $2x^3 - 8x^2 + 5x - 7$ and the remainder is $8$.

Conclusion

In this article, we have shown how to divide the polynomial $2x^4 - 4x^3 - 11x^2 + 3x - 6$ by $x + 2$ using the long division method. We have performed the division step by step and obtained the quotient and remainder. The quotient is $2x^3 - 8x^2 + 5x - 7$ and the remainder is $8$.
Divide $2x^4 - 4x^3 - 11x^2 + 3x - 6$ by $x + 2$: Q&A

Q: What is the long division method for polynomials?

A: The long division method for polynomials is a process of dividing one polynomial by another. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the divisor by the quotient and subtracting the result from the dividend.

Q: How do I set up the division problem?

A: To set up the division problem, you will write the dividend and the divisor in the following format:

\begin{array}{rrrrr} 2 & -4 & -11 & 3 & -6 \ & -4 & 16 & -10 & 14 \ \hline 2 & -8 & 5 & -7 & 8 \end{array}

Q: What is the first step in the long division method?

A: The first step in the long division method is to divide the highest degree term of the dividend by the highest degree term of the divisor. In this case, the highest degree term of the dividend is $2x^4$ and the highest degree term of the divisor is $x$. We will divide $2x^4$ by $x$ to get $2x^3$.

Q: How do I perform the division?

A: To perform the division, you will multiply the divisor by the quotient and subtract the result from the dividend. You will repeat this process until you have no more terms to divide.

Q: What is the remainder in the long division method?

A: The remainder in the long division method is the result of the final subtraction. In this case, the remainder is $8$.

Q: How do I determine the quotient and remainder?

A: To determine the quotient and remainder, you will look at the final result of the long division method. The quotient will be the result of the division, and the remainder will be the result of the final subtraction.

Q: What is the significance of the quotient and remainder?

A: The quotient and remainder are important in the long division method because they tell us how many times the divisor fits into the dividend, and what is left over.

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

A: Yes, you can use the long division method for any polynomial division. However, the method may become more complicated for polynomials of higher degree.

Q: Are there any shortcuts for the long division method?

A: Yes, there are shortcuts for the long division method. One shortcut is to use synthetic division, which is a faster and more efficient method for dividing polynomials.

Q: How do I check my work in the long division method?

A: To check your work in the long division method, you will multiply the divisor by the quotient and add the remainder. If the result is equal to the dividend, then your work is correct.

Q: What are some common mistakes to avoid in the long division method?

A: Some common mistakes to avoid in the long division method include:

• Not performing the division correctly
• Not subtracting the result of the multiplication from the dividend
• Not checking the work
• Not using the correct order of operations

Q: Can I use the long division method for polynomial division with complex numbers?

A: Yes, you can use the long division method for polynomial division with complex numbers. However, the method may become more complicated due to the presence of complex numbers.

Q: Are there any other methods for polynomial division?

A: Yes, there are other methods for polynomial division, including synthetic division and polynomial long division. Each method has its own advantages and disadvantages, and the choice of method will depend on the specific problem and the level of difficulty.