Chapter A.4, Problem 8AYU

To determine

To calculate: The quotient and remainder when the polynomial $-4x^3+2x^2-x+1$ is divided by $x+2$ with help of synthetic division.

Answer to Problem 8AYU

The quotient is $-4x^2+10x-21$ and reminder is $43$ .

Explanation of Solution

Given information:

The polynomial $-4x^3+2x^2-x+1$ and a factor of it $x+2$ .

Formula used:

When a polynomial is divided by its factor then dividend is the product of divisor and quotient increased by remainder.

Calculation:

Consider the provided polynomial $-4x^3+2x^2-x+1$ and a factor of it $x+2$ .

To divide the polynomial $-4x^3+2x^2-x+1$ by $x+2$ follow the steps provided below,

Here dividend is $-4x^3+2x^2-x+1$ and divisor is $x-\left(-2\right)$ .

Step 1: List down the coefficients of dividend in the descending powers of x, that are $-4,2,-1,1$ .

Now, the divisor is of the form $x-c$ , where $c=-2$ .

Step 2: To perform the synthetic division put the coefficients in the division sign along with $-2$ on the left side,

  $-2\overline{)\begin{array}{cccc}-4& 2& -1& 1\end{array}}$

Step 3: Bring down $-4$ to in third row,

  $-2\overline{)\begin{array}{l}\begin{array}{cccc}-4& 2& -1& 1\end{array}\\ \begin{array}{cccc}& & & \end{array}\\ -4\end{array}}$

Step 4: Multiply $-2$ with the first entry of row 3 and place the result in row 2 column 2.

  $-2\overline{)\begin{array}{l}\begin{array}{cccc}-4& 2& -1& 1\end{array}\\ \begin{array}{cccc}& 8& & \end{array}\\ \begin{array}{cccc}-4& & & \end{array}\end{array}}$

Step 5: Now, add the elements of column 2 of row 1 and row 2.

  $-2\overline{)\begin{array}{l}\begin{array}{cccc}-4& 2& -1& 1\end{array}\\ \begin{array}{cccc}& 8& & \end{array}\\ \begin{array}{cccc}-4& 10& & \end{array}\end{array}}$

Step 6: Multiply $-2$ with the second entry of row 3 and place the result in row 2 column 3.

  $-2\overline{)\begin{array}{l}\begin{array}{cccc}-4& 2& -1& 1\end{array}\\ \begin{array}{cccc}& 8& -20& \end{array}\\ \begin{array}{cccc}-4& 10& & \end{array}\end{array}}$

Step 7: Now, add the elements of column 3 of row 1 and row 2.

  $-2\overline{)\begin{array}{l}\begin{array}{cccc}-4& 2& -1& 1\end{array}\\ \begin{array}{cccc}& 8& -20& \end{array}\\ \begin{array}{cccc}-4& 10& -21& \end{array}\end{array}}$

Step 8: Multiply $-2$ with the third entry of row 3 and place the result in row 2 column 4.

  $-2\overline{)\begin{array}{l}\begin{array}{cccc}-4& 2& -1& 1\end{array}\\ \begin{array}{cccc}& 8& -20& 42\end{array}\\ \begin{array}{cccc}-4& 10& -21& \end{array}\end{array}}$

Step 9: Now, add the elements of column 4 of row 1 and row 2.

  $-2\overline{)\begin{array}{l}\begin{array}{cccc}-4& 2& -1& 1\end{array}\\ \begin{array}{cccc}& 8& -20& 42\end{array}\\ \begin{array}{cccc}-4& 10& -21& 43\end{array}\end{array}}$

Now, the first three elements of row 3 are coefficients of quotient in descending powers of x with degree one less than dividend and the last entry is remainder.

Therefore, quotient is $-4x^2+10x-21$ and reminder is 43.

Now, when a polynomial is divided by its divisor then dividend is the product of divisor and quotient increased by remainder.

  $-4x^3+2x^2-x+1=\left(-4x^2+10x-21\right)\left(x+2\right)+43$

To verify multiply the terms on the right hand side,

  $\begin{array}{c}\left(-4x^2+10x-21\right)\left(x+2\right)+43=-4x^3-8x^2+10x^2+20x-21x-42+43\\ =-4x^3+2x^2-x+1\end{array}$

Thus, the quotient is $-4x^2+10x-21$ and reminder is $43$ when the polynomial $-4x^3+2x^2-x+1$ is divided by $x+2$ .