How do you divide #(3x^3+4x^2+x+1)÷(x-1)#?

Question

Nathan Axel · Accepted Answer

#3x^2+7x+8+(9)/(x-1)#

You can use Remainder Theorem or Synthetic Division to do so. I will use the Synthetic Division since it is quicker and easier (but it only works for a polynomial divided by a binomial).

The first step in using theSynthetic Division is to find out what makes the binomial #0#. In this case, it is #1#. Now, you put that number in a sort of secluded area like so:

#1" "|#

Then, you determine the co-efficients for every term in your polynomial (you MUST also include co-efficients of #0#. However, there are none in this problem). Then, you make a list like this:

#1" "|color(white)(xx)3" " "4" " "1color(white)(xx)1#

Then, you do this little trick where you bring down the first number (#3#), multiply by the secluded number (#1#), put the answer down under the next number (#4#), and then add. After its all done, it'll look like this:

#1" "|color(white)(xx)3" " "4" " "1color(white)(xx)1# #color(white)(xxxxxx)darrcolor(white)(x)3" " "7color(white)(xx)8# #color(white)(xxxxx)stackrel("---------------------------------")# #color(white)(xxxxxx)3" " "7" " "8|color(white)(x)9#

Now, the remainder has the co-efficients on the third line (#3#, #7#, and #8#), making the polynomial #3x^2+7x+8#. The #9# is the remainder (divided by the binomial).

All-in-all, the answer is:

#3x^2+7x+8+(9)/(x-1)#.

I know the description was confusing, which is why I have attached a link to a video that will do a better job explaining it to you!

Abigail Allen · Answer

undefined To divide (3x^3+4x^2+x+1) by (x-1), you can use polynomial long division. Here are the steps:

1. Divide the first term of the dividend (3x^3) by the first term of the divisor (x). This gives you 3x^2.
2. Multiply the entire divisor (x-1) by the quotient obtained in step 1 (3x^2). This gives you 3x^3-3x^2.
3. Subtract the result obtained in step 2 from the original dividend (3x^3+4x^2+x+1 - 3x^3+3x^2). This simplifies to x^2+x+1.
4. Bring down the next term from the dividend, which is x^2. Now you have x^2+x+1 as the new dividend.
5. Repeat steps 1-4 until you have no more terms to bring down.

Continuing with the steps:
6. Divide x^2 by x, which gives you x.
7. Multiply the entire divisor (x-1) by the quotient obtained in step 6 (x). This gives you x^2-x.
8. Subtract the result obtained in step 7 from the current dividend (x^2+x+1 - x^2+x). This simplifies to 1.
9. Bring down the next term from the dividend, which is 1. Now you have 1 as the new dividend.
10. Divide 1 by x, which gives you 0.
11. Multiply the entire divisor (x-1) by the quotient obtained in step 10 (0). This gives you 0.
12. Subtract the result obtained in step 11 from the current dividend (1 - 0). This simplifies to 1.

Therefore, the quotient of (3x^3+4x^2+x+1) divided by (x-1) is 3x^2 + x + 1, with a remainder of 1.