Synthetic Division

Study Notes

Definition: Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form (x - a) and finding the remainder.

Steps:

Write the coefficients of the dividend (the polynomial being divided) in a row, with the coefficient of the leading term first.
Write the zero of the divisor (the binomial) below the line.
Bring down the leading coefficient.
Multiply the number below the line by the leading coefficient and subtract the product from the next coefficient.
Repeat step 4 until the last coefficient is reached.
The final result is the quotient and the remainder.

Example: Divide x^3 + 2x^2 - 7x - 12 by x - 2 using synthetic division.

   2 | 1   2  -7  -12
      | 2   8   2
      -------------------
         1   4   1  -10

Quotient: x^2 + 4x + 1 Remainder: -10

Properties:

If the remainder is zero, then (x - a) is a factor of the polynomial.
Synthetic division can be used to find the zeros of a polynomial.
It can also be used to test whether a given value of x is a zero of the polynomial.

Synthetic division is a method used for dividing polynomials by binomials in the form of (x - a).
It provides a systematic approach to find both the quotient and the remainder.

Start by listing the coefficients of the polynomial in a row, beginning with the leading term.
Identify the zero of the divisor (x - a) and place it below the division line.
The leading coefficient is brought down first to begin the process.
Multiply the zero by the leading coefficient, then subtract this from the subsequent coefficient.
Continue this multiplication and subtraction until all coefficients are processed.
The result at the end shows the quotient and the remainder from the division.

For the polynomial x^3 + 2x^2 - 7x - 12 divided by x - 2:

   2 | 1   2  -7  -12
      | 2   8   2
      -------------------
         1   4   1  -10

A remainder of zero indicates that (x - a) is a factor of the polynomial.
Synthetic division can also assist in finding the zeros of a polynomial.
It is useful for verifying if a specific value of x is a zero of the polynomial.