Dividing Polynomials

Remainder and Factor Theorems
Long Division of Polynomials
1.Arrange the terms of both the dividend and the divisor in descending powers of any variable .
2.Divide the first term in the dividend by the first term in the divisor. The result is the first term of the quotient.
3.Multiply every term in the divisor by the first term in the quotient. Write the resulting product beneath the dividend with like terms lined up .
4. Subtract the product from the dividend.
5.Bring down the next term in the original dividend and write it next to the remainder to form a new dividend.
6.Use this new expression as the dividend and repeat this process until the remainder can no longer be divided. This will occur when the degree of the remainder is less than the degree of the divisor.

The Division Algorithm
If f(x) and d(x) are polynomials, with d(x)≠0,and the degree of d(x) is less than or equal tothe degree of f(x), then there exist unique polynomials q (x) and r(x) such that

The Remainder
The remainder, r(x), equals 0 or it is of degree less than the degree of d(x).If r(x) 0, we say that d(x) divides evenly into f(x) and that d(x) and q(x) are factors of f(x).

Synthetic Division
To divide a polynomial by .x−c.
1 Arrange polynomials in descending powers, with a 0 coefficient for any missing term.
Example

Synthetic Division contd.

2. Write for the divisor, x-c. To the right, write the coefficients of the dividend

Example

3. Write the leading coefficient of the dividend on the bottom row.
Example

4. Multiply times the value just written on the bottom row. Write the product in the next column in the second row.
Example

5. Add the values in this new column, wirting the sum in the bottom row.
Example
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6. Repeat this series of multiplications and additions until all columns are filled in.
Example

7. Use the numbers in the last row to write the quotient and the remainder in f tractional form The degree of the first term the quotient is one less than the degree of the first term of the dividend.  The final valuein in this row is the remainder.

Example
Use synthetic division to divide
Solution

The Remainder Theorem
If the polynomial f(x) is divided by x-c, then the remainder is f(c).

The Factor Theorem
Let f(f) be a polynomial.
a. If f(x)=0, then is a factor of f(x).
b. If x-c is a factor of f(x), then f(x)=0.