Synthetic division

Synthetic Division Method: Synthetic division is a method for dividing a polynomial by a linear factor of the form $(x-a)$. This method is often used to find the quotient and remainder of the division quickly and easily.

To perform synthetic division, we write the coefficients of the polynomial in a row and then perform a series of operations involving the divisor and the coefficients to obtain the quotient and remainder.

Here are the steps for synthetic division (with the help of an example):

Suppose we want to divide the polynomial $p(x) = 3x^4 - 2x^3 + 5x - 6$ by the factor $(x-2)$

1. Write the coefficients of the polynomial in a row, with a placeholder for any missing terms.

   

2. Write the value of the divisor, with the opposite sign, to the left of the row of coefficients.

   

3. Bring down the first coefficient.

   

4. Multiply the value of the divisor by the coefficient just brought down, and write the result under the next coefficient.

   

5. Add the two values just written down to get a new value, and write it under the horizontal line.

   

6. Repeat steps 4 and 5 for each coefficient in the row.

   

7. The last value written down below the line is the remainder, and the values above the line form the coefficients of the quotient.

   

8. In the last row, the last digit is the remainder and the first four numbers are the coefficient of the quotient when the given polynomial is divided by $x-2$. i.e.

   Remainder $= 36$

   Quotient: $3x^3 + 4x^2 + 8x + 21$

Synthetic division is useful in several ways:

1. It provides a quick and easy way to divide a polynomial by a linear factor of the form $(x-a)$, which provides the quotient and remainder of the division.

2. It can be used to find the value of unknown coefficients if zeros of the polynomials are given.

3. It can be used to factor polynomials, by identifying factors that produce zero remainders when dividing by a linear factor.

Example: For what value of $m$, $1$ is a zero of the polynomial $p(x) = x^3 - mx^2 + x - 1$.

Solution:

We can use synthetic division to evaluate the polynomial $p(x)$ at $x=1$ and see if the remainder is zero. If the remainder is zero, then $x=1$ is a zero of the polynomial.

The last entry in the bottom row is the remainder, which is $1-m$. For $x=1$ to be a zero of the polynomial, the remainder must be zero. Therefore, we need to solve the equation $1-m = 0$, which implies that $m=1$.

Hence, if $m=1$, then $x=1$ is a zero of the polynomial $p(x)$.