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## Algebra (all content)

### Course: Algebra (all content)>Unit 10

Lesson 22: Synthetic division of polynomials

# Dividing polynomials: synthetic division

Sal divides (2x^5-x^3+3x^2-2x+7) by (x-3) using synthetic division. Created by Sal Khan.

## Want to join the conversation?

• What if there isn't a remainder? • I'm trying to find a video of yours to help me figure out how to do these math problems that involve synthetic division but they also factoring. She told us, " use synthetic division to show that X is a solution of the third degree polynomial equation and use the result to factor completely. List all real solutions". I'm not real sure how to do all this, i've never been very good at factoring anyway... help? •  Let's say you have a 3rd degree polynomial p(x). If R is a root of p(x), the monomial (x - R) divides p(x) and there is no remainder.

After you've divided p(x) with (x - R) and thus proven that R is a root, you should have a quadratic equation which you can probably factor on your own.

Example: x = 1 is a solution to p(x) = 4x^3 - 8x^2 - 20x + 24 = 0. Factor completely and list all real solutions.

Step 1: Divide p(x) with (x - 1): (4x^3 - 8x^2 - 20x + 24) / (x - 1) = 4x^2 - 4x - 24.
There's no remainder, so x = 1 is indeed a root of p(x).

Step 2. Factor what we got in step 1: 4x^2 - 4x - 24. You can factor it by solving its roots with the quadratic formula, or whichever way you want to do it.

The roots of 4x^2 - 4x - 24 are x = -2 and x = 3, so it factors to 4(x + 2)(x - 3).

So the original polynomial factors to 4(x + 2)(x - 3)(x - 1) and its real roots are x = -2, x = 1 and x=3.
• How did the first term of 2x^5 go to 2x^4? • He says there is a different way to do it if you are not dividing by x + or - a number. Are there Videos on this? • Does anyone know how/who synthetic divison was thought of/who though of it? And also, where did the "operation" symbol come from? • The "operation symbol" is not actually an operator, it's just a template to organize the calculations for this particular operation, similar to long division or multiplication by hand. You could think of other "operators", but this one is conventional. As for your other questions, Google is your friend :)
• Is there a video on how to do synthetic division if the number being divided by starts with x squared? • What about in the case of the polynomial having more than 1 variable. In one question, there was a polynomial; 9a^2 - 30ab + 25b^2 Divided by 3a - 5b, What do I do then? • it would be easier to do polynomial long division because this synthetic division only works when the denominator / devisor is x+number or x-number, with a coefficient of 1. at , sal says this.

with a coefficient greater or less than one connected to the x of denominator, you would divide the denominator and polynomial with that coefficient and solve using your new polynomial and denominator (when the coefficient of x is 1), in other words multiplying 1/coefficient
(1 vote)
• When we finally write the answer as 17x^2 + 54x + 160 + 487/x-3 , why do we write 487/x-3 and not just 487?   