Chapter A.4, Problem 7AYU

To determine

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

Answer to Problem 7AYU

The quotient is $3x^2+11x+32$ and reminder is $99$ .

Explanation of Solution

Given information:

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

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 $3x^3+2x^2-x+3$ and a factor of it $x-3$ .

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

Here dividend is $3x^3+2x^2-x+3$ and divisor is $x-3$ .

Step 1: List down the coefficients of dividend in the descending powers of x, that are $3,2,-1,3$ .

Now, the divisor is of the form $x-c$ , where $c=3$ .

Step 2: To perform the synthetic division put the coefficients in the division sign along with 3 on the left side,

  $3\overline{)\begin{array}{cccc}3& 2& -1& 3\end{array}}$

Step 3: Bring down 3 to in third row,

  $3\overline{)\begin{array}{l}\begin{array}{cccc}3& 2& -1& 3\end{array}\\ \begin{array}{cccc}& & & \end{array}\\ \begin{array}{cccc}3& & & \end{array}\end{array}}$

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

  $3\overline{)\begin{array}{l}\begin{array}{cccc}3& 2& -1& 3\end{array}\\ \begin{array}{cccc}& 9& & \end{array}\\ \begin{array}{cccc}3& & & \end{array}\end{array}}$

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

  $3\overline{)\begin{array}{l}\begin{array}{cccc}3& 2& -1& 3\end{array}\\ \begin{array}{cccc}& 9& & \end{array}\\ \begin{array}{cccc}3& 11& & \end{array}\end{array}}$

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

  $3\overline{)\begin{array}{l}\begin{array}{cccc}3& 2& -1& 3\end{array}\\ \begin{array}{cccc}& 9& 33& \end{array}\\ \begin{array}{cccc}3& 11& & \end{array}\end{array}}$

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

  $3\overline{)\begin{array}{l}\begin{array}{cccc}3& 2& -1& 3\end{array}\\ \begin{array}{cccc}& 9& 33& \end{array}\\ \begin{array}{cccc}3& 11& 32& \end{array}\end{array}}$

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

  $3\overline{)\begin{array}{l}\begin{array}{cccc}3& 2& -1& 3\end{array}\\ \begin{array}{cccc}& 9& 33& 96\end{array}\\ \begin{array}{cccc}3& 11& 32& \end{array}\end{array}}$

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

  $3\overline{)\begin{array}{l}\begin{array}{cccc}3& 2& -1& 3\end{array}\\ \begin{array}{cccc}& 9& 33& 96\end{array}\\ \begin{array}{cccc}3& 11& 32& 99\end{array}\end{array}}$

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 $3x^2+11x+32$ and reminder is 99.

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

  $3x^3+2x^2-x+3=\left(3x^2+11x+32\right)\left(x-3\right)+99$

To verify multiply the terms on the right hand side,

  $\begin{array}{c}\left(3x^2+11x+32\right)\left(x-3\right)+99=3x^3-9x^2+11x^2-33x+32x-96+99\\ =3x^2+2x^2-x+3\end{array}$

Thus, the quotient is $3x^2+11x+32$ and reminder is $99$ when the polynomial $3x^3+2x^2-x+3$ is divided by $x-3$ .