Polynomial Division
Divide (3x^4 - 2x^3 + 4x^2 - x - 4) by (x - 1) using synthetic division
3x^3 + x^2 + 5x + 4
Divide (3x^4 + 6x^3 - 9x^2 + 2x + 6) by (x + 3) using long division
3x^3 - 3x^2 + 2
Divide (4x^4 + 9x^3 - 2x^2 + 7x + 5) by (4x + 1) using synthetic division
4x^3 + 8x^2 - 4x + 8 R3
Divide (4x^4 + 2x^3 - 7x^2 + 6x - 5) by (x + 2) using long division
4x^3 - 6x^2 + 5x - 4 + [3 / (x + 2)]
Divide (4x^4 - 9x^3 - 8x^2 + 2x + 25) by (x - 2) using long division
4x^3 - x^2 - 10x - 18 - [11 / (x - 2)]
Divide (5x^3 - 3x^2 -6) by (x-1) using synthetic division
5x^2 + 2x + 2 R-4
Divide (6x^4 + 7x^3 - 8x^2 + 9x + 10) by (2x + 1) using synthetic division
6x^3 + 4x^2 - 10x + 14 R3
Divide (8x^3 - 6x^2 + 4x - 6) by (x - 1) using synthetic division
8x^2 + 2x + 6
Divide (9x^4 - 6x^3 + 4x^2 - 7x + 2) by (3x - 1) using synthetic division
9x^3 - 3x^2 + 3x - 6 = 3x^3 - x^2 + x - 2
Divide (x^3 + x^2 - 40x - 450) by (x - 9) using long division
x^2 + 10x + 50
Divide (x^3 + 3x^2 + 4x - 8) by (x - 1) using long division
x^2 + 4x + 8
Divide (x^3 + 8x^2 + 10x - 4) by (x + 2) using long division
x^2 + 6x - 2
Divide (x^3 + 3x^2 - 7x + 12) by (x - 4) using long division
x^2 + 7x + 21 + [96 / (x - 4)]
Divide (x^3 - 5x^2 + 10x - 18) by (x - 4) using synthetic division
x^2 - 2 + 6 R6
Divide (x^3 - 2x^2 - 10x + 25) by (x + 5) using long division
x^2 - 7x + 25 - [100 / (x + 5)]
Divide (x^3 - 5x^2 + 6x - 8) by (x - 4) using synthetic division
x^2 - x + 2
Divide (3x^4 + 9x^3 + 8x^2 + 5x + 3) by (x + 1) using long division
3x^3 + 6x^2 + 2x + 3
Steps for using Long Division
1. Divide the first term of the numerator by the first term of the denominator. 2. Multiply the denominator by the term found above. 3. Subtract this from the numerator. 4. Repeat these steps until every term has been brought down.
Steps for using Synthetic Division
1. First set up the problem: set the denominator equal to 0 and solve to find the number that goes in the division box, then list the coefficient of each term in the numerator. 2. Bring the leading coefficient down and multiply it by the term in the division box. 3. Write the number (calculated in the step above) in the next column, then add the two numbers in the column and bring the result down to the bottom. 4. Repeat these steps until you get to the end of the problem.
Divide (10x^4 - 3x^3 - 9x^2 + 14x - 5) by (5x - 4) using synthetic division
10x^3 + 5x^2 - 5x + 10 R3
Divide (12x^4 + 5x^3 - 7x^2 + 9x + 9) by (4x + 3) using synthetic division
12x^3 - 4x^2 -4x + 12 = 3x^3 - x^2 - x + 3
Divide (2x^3 + 10x^2 - 20x - 40) by (x + 6) using synthetic division
2x^2 - 2x - 8 R8
Divide (2x^3 + 7x^2 - 3x + 15) by (x + 5) using synthetic division
2x^2 - 3x + 12 R-45
Divide (2x^3 + 7x^2 -6x - 8) by (x + 4) using long division
2x^2 - x - 2
Divide (2x^4 - 5x^3 - 7x^2 + 8x + 15) by (x - 3) using synthetic division
2x^3 + x^2 - 4x - 4 R3
Divide (2x^4 + 5x^3 - 10x^2 - 7x - 35) by (x + 4) using long division
2x^3 - 3x^2 + 2x - 15 + [25 / (x + 4)]
Divide (2x^4 + 5x^3 - 5x^2 - 3x + 8) by (x + 3) using long division
2x^3 - x^2 - 2x + 3 - [1 / (x + 3)]
Divide (3x^3 + 10x^2 + 5x - 6) by (x + 2) using synthetic division
3x^2 + 4x - 3
Divide (3x^3 - 2x^2 - 7x + 6) by (x + 1) using long division
3x^2 - 5x - 2 + [8 / (x + 1)]
Divide (3x^3 - 5x^2 + 4x + 2) by (3x + 1) using long division
3x^2 - 6x + 6
Divide (3x^4 - 5x^2 - x + 3) by (x + 2) using long division
3x^2 - 6x^2 + 7x - 15 + [33 / (x + 2)]
Divide (3x^4 + 10x^3 + 7x^2 - 4x - 4) by (3x - 2) using synthetic division
3x^3 + 12x^2 + 15x + 6
Missing Terms
When doing polynomial division on a polynomial that is missing one or more terms, you must include the missing terms but with a coefficient of 0, for example (x^3 + 4x - 7) must be written as (x^3 + 0x^2 + 4x - 7)
Dividing a polynomial without a given factor
When you are trying to divide and factor a polynomial without a given factor, you can test different numbers by plugging them in as factors and completing polynomial division. You can use the rational root theorem to find all possible rational roots of the polynomial, then test the roots from this list. If there is no remainder the number you chose is a root; if there is a remainder this number is not a root and you must try a new one.
Finding Factors
You know that you've found a factor of a polynomial if you perform long or synthetic division and you end up with a remainder of 0
