Chapter A.4, Problem 28AYU

To determine

To explain: The better method among long division and synthetic division when a polynomial is divided by $x-c$ and value of c in deciding in method.

Answer to Problem 28AYU

The better method to divide the polynomials when divided by $x-c$ is the synthetic division. It does not depend on value of c.

Explanation of Solution

Given information:

The two methods to divided a polynomial by $x-c$ are long division and synthetic division.

To divide a polynomial by linear polynomial whose leading coefficient is 1 and is of the form $x-c$ where c is a real number, synthetic division is preferred over long division. Synthetic division is a shortcut method to long division and is easily understandable.

Synthetic division is the compact form of long division and is much simpler in terms of notation.

The value of c does not change the decision of choosing the better method of dividing the polynomials.

For example: consider the provided polynomial $x^4+x^2+1$ and a factor of it $x-2$ .

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

Here dividend is $x^4+0\cdot x^3+x^2+0\cdot x+1$ and divisor is $x-2$ .

Step 1: List down the coefficients of dividend in the descending powers of x, that are $1,0,1,0,1$ .

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,

  $2\overline{)\begin{array}{ccccc}1& 0& 1& 0& 1\end{array}}$

Step 3: Bring down $1$ to in third row,

  $2\overline{)\begin{array}{l}\begin{array}{ccccc}1& 0& 1& 0& 1\end{array}\\ \begin{array}{ccccc}& & & & \end{array}\\ \begin{array}{ccccc}1& & & & \end{array}\end{array}}$

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

  $2\overline{)\begin{array}{l}\begin{array}{ccccc}1& 0& 1& 0& 1\end{array}\\ \begin{array}{ccccc}& 2& & & \end{array}\\ \begin{array}{ccccc}1& & & & \end{array}\end{array}}$

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

  $2\overline{)\begin{array}{l}\begin{array}{ccccc}1& 0& 1& 0& 1\end{array}\\ \begin{array}{ccccc}& 2& & & \end{array}\\ \begin{array}{ccccc}1& 2& & & \end{array}\end{array}}$

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

  $2\overline{)\begin{array}{l}\begin{array}{ccccc}1& 0& 1& 0& 1\end{array}\\ \begin{array}{ccccc}& 2& 4& & \end{array}\\ \begin{array}{ccccc}1& 2& & & \end{array}\end{array}}$

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

  $2\overline{)\begin{array}{l}\begin{array}{ccccc}1& 0& 1& 0& 1\end{array}\\ \begin{array}{ccccc}& 2& 4& & \end{array}\\ \begin{array}{ccccc}1& 2& 5& & \end{array}\end{array}}$

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

  $2\overline{)\begin{array}{l}\begin{array}{ccccc}1& 0& 1& 0& 1\end{array}\\ \begin{array}{ccccc}& 2& 4& 10& \end{array}\\ \begin{array}{ccccc}1& 2& 5& & \end{array}\end{array}}$

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

  $2\overline{)\begin{array}{l}\begin{array}{ccccc}1& 0& 1& 0& 1\end{array}\\ \begin{array}{ccccc}& 2& 4& 10& \end{array}\\ \begin{array}{ccccc}1& 2& 5& 10& \end{array}\end{array}}$

Repeat the above steps 8 and 9 for remaining elements of first row,

  $2\overline{)\begin{array}{l}\begin{array}{ccccc}1& 0& 1& 0& 1\end{array}\\ \begin{array}{ccccc}& 2& 4& 10& 20\end{array}\\ \begin{array}{ccccc}1& 2& 5& 10& 21\end{array}\end{array}}$

Now, the first four 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 $x^3+2x^2+5x+10$ and reminder is $21$ .

Thus, the better method to divide the polynomials when divided by $x-c$ is the synthetic division. It does not depend on value of c.