Synthetic Division Calculator

Publish date: 2024-08-03

Use synthetic division for:

3x4 + 6x3 - 123x2 - 126x + 1080
x - 3

Determine our root divisor:

Solve the divisor equation x - 3 = 0
Add 3 to each side of the equation to get
x - 3 + 3 = 0 + 3
Therefore, our root becomes x = 3

	3	6	-123	-126	1080
3

	3	6	-123	-126	1080
3	↓
	3

Multiply our root 3 by 3
Take 9 and put that in column 2:

	3	6	-123	-126	1080
3		9
	3

9 + 6 = 15
Put this in the answer column 2:

	3	6	-123	-126	1080
3		9
	3	15

Multiply our root 3 by 15
Take 45 and put that in column 3:

	3	6	-123	-126	1080
3		9	45
	3	15

45 + -123 = -78
Put this in the answer column 3:

	3	6	-123	-126	1080
3		9	45
	3	15	-78

Multiply our root 3 by -78
Take -234 and put that in column 4:

	3	6	-123	-126	1080
3		9	45	-234
	3	15	-78

-234 + -126 = -360
Put this in the answer column 4:

	3	6	-123	-126	1080
3		9	45	-234
	3	15	-78	-360

Multiply our root 3 by -360
Take -1080 and put that in column 5:

	3	6	-123	-126	1080
3		9	45	-234	-1080
	3	15	-78	-360

-1080 + 1080 = 0
Put this in the answer column 5:

	3	6	-123	-126	1080
3		9	45	-234	-1080
	3	15	-78	-360	0

Leading Answer Term = x(4 - 1) = x3

Since the last number in our result line = 0, we will not have a remainder and have a clean quotient which is shown below in our answer:

Answer = 3x3 + 15x2 - 78x - 360

It appears your answer forms a cubic equation since the maximum power of your result equation is 3 and your remainder is zero. Click here to solve this cubic equation

How does the Synthetic Division Calculator work?

Free Synthetic Division Calculator - Using Ruffinis Rule, this performs synthetic division by dividing a polynomial with a maximum degree of 6 by a term (x ± c) where c is a constant root using the factor theorem. The calculator returns a quotient answer that includes a remainder if applicable. Also known as the Rational Zero Theorem
This calculator has 1 input.

What 3 formulas are used for the Synthetic Division Calculator?

For x + a, our root is -1 * a
Write down our coefficients horizontally and our root of a to the left
Multiply root by last result and repeat process

For more math formulas, check out our Formula Dossier

What 4 concepts are covered in the Synthetic Division Calculator?

divisionseparate a number into partspolynomialan expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).ruffinis rulean efficient method of dividing a polynomial by a linear factor, with or without a remainder. synthetic divisiona shorthand method for dividing a polynomial by a linear factor