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## Algebra (all content)

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

Lesson 23: Practice dividing polynomials with remainders

# Divide polynomials by x (with remainders)

Learn how to simplify complex expressions by dividing polynomials by 'x'. Discover how to break down the numerator, distribute the division, and use exponent properties. Master the art of simplifying fractions and handling negative exponents. Sal demonstrates by dividing (18x^4-3x^2+6x-4) by 6x. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Shouldn't the two fractions at the end of the video have been added together to make a single fraction?
• Bobby The 1/2 times x has an x to the positive one power and the 2/3 x has a negative one power there for they do not qualify as like terms. Because the first x is the same as X/1 and the second is 1/x. Its the same as the difference between 100 and 1/100 :D Hope this helps
• Could you use synthetic division to solve this problem?
If yes, how?
• No, you cannot. In order to divide polynomials using synthetic division, the denominator (the number(s) on the bottom of the fraction) must satisfy two rules:
1 - Be a linear expression, in other words, each term must either be a constant or the product of a constant and a single variable to the power of 1.
2 - The leading coefficient (first number) must be a 1.
For example, you can use synthetic division to divide a polynomial by (x + 2) or (x – 6), but you cannot use synthetic division to divide by 6x, or (2x + 3) or (3x^2 – x + 3).
• At , Sal said, "anything divided by anything is just one." I think what he meant to say was "anything divided by itself is just one." :)
• Yeah,
Sal was referring to the first anything when he said anything the second time : )
• At the End of the video, he said we can write it as x^-1 power, but by the way of the polynomials it is not a polynomial because every degree in the expression must have a non-negative integer, so what's with that Am I wrong or I am missing something.
• Dividing 2 polynomials doesn't guarantee that your result will be a polynomial. This is just like if we divide 2 whole numbers, our result may or may not ge a whole number. For example: 17 divided by 5 = 3 2/5 or 3.4. The result is not a whole number. We get a rational number (one involving a fraction).

Sal's result is a rational expression (one that involves a fraction).

Hope this helps.
• what if instead of 6x on the bottom as the denominator its negative 6x how would you distribute it when doing the division
• Then you would simplify your equation by distributing the negative from the denominator (in your case the 6x) throughout its numerator.
So using Mr. Khan's example of the numerator being: 18x^4-3x^2+6x-4... and dividing it by its new denominator of -6x (your question), then you would multiply both the numerator and the denominator by -1, which would simplify the equation for you by making it a negative numerator divided by a positive denominator like this:
Numerator... -1(18x^4-3x^2+6x-4)= -18x^4+3x^2-6x+4.
& Denominator... -1(-6x)= +6x.

Hope that helped!
• At to , could the `-½x` also be written as `-x/2`?
• yes they can be because they are similar expressions
(1 vote)
• I heard in some videos when the power of x is raised to negative exponent it is not consider as polynomial or is this just some convention?
• That is correct, it is not a polynomial. By definition, the exponents of all the x's must be a nonnegative integer. If the exponent is negative or not an integer, then it is not a polynomial.

There are some more advanced definitions of "polynomial" that professional mathematicians work with, but for this level of study we can define a polynomial as a function that can be written (that doesn't mean it is currently written this way, but only that it is possible) as the sum of a finite number of terms that are composed of a constant multiplied by a variable raised to a nonnegative integer power. Note that some of these terms can have the constant multiplied by the variable raised to the zeroth power which would be equal to just the constant.
• when divide a polynomial why do we get a reminder that is 1 power less x^n than the divisor?
• At , Sal writes it as x^1 but it's just a remainder.
• Do you add or subtract the exponents?