Chapter A.4, Problem 1AYU

To determine

To fill: The blank in the statement “To check division, the expression that is being divided, the dividend, should equal to product of the _________ and ___________ plus the _________.’’

Answer to Problem 1AYU

The complete statement is “To check division, the expression that is being divided, the dividend, should equal to product of the divisor and quotient plus theremainder.”

Explanation of Solution

Given information:

The statement “To check division, the expression that is being divided, the dividend, should equal to product of the __________ and ___________ plus the ___________.”

Consider the provided statement “To check division, the expression that is being divided, the dividend, should equal to product of the ___________ and ___________ plus the ___________.”

The dividend is the product of divisor and quotient added to remainder.

For example: consider the provided polynomial $3x^3+2x^2-x+2$ and a factor of it $x-3$ .

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

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

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

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,

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

Step 3: Bring down 3 to in third row,

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

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

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

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

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

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

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

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

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

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

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

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

  $3\overline{)\begin{array}{l}\begin{array}{cccc}3& 2& -1& 2\end{array}\\ \begin{array}{cccc}& 9& 33& 96\end{array}\\ \begin{array}{cccc}3& 11& 32& 98\end{array}\end{array}}$

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 98.

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

  $3x^3+2x^2-x+2=\left(3x^2+11x+32\right)\left(x-3\right)+98$

To verify multiply the terms on the right hand side,

  $\begin{array}{c}\left(3x^2+11x+32\right)\left(x-3\right)+98=3x^3-9x^2+11x^2-33x+32x-96+98\\ =3x^2+2x^2-x+2\end{array}$

Thus, the complete statement is “To check division, the expression that is being divided, the dividend, should equal to product of the divisor and quotient plus theremainder.”