Find the Quotient and remainder using long division.
1. (x^3+1) / (x^2+x+2)
2. (14x^3-16x^2+10x+8) / (x-1)
Please explain because I want to learn how to do this correctly. Thanks
To do that for you would require a lot of graphics that are difficult to type here with the proper alignment.
Rather than explain to you how to solve those problems with polynomial long division, I think it is better to give you a website or two that will show you how to do it yourself.
Here is a good one:
http://www.purplemath.com/modules/polydiv2.htm
If that is hard to follow, Google "polynomial long division" to get other good tutorial sites.
If you have to deal with remainders, this is a better site:
://www.sosmath.com/algebra/factor/fac01/fac01.html
Sure! I'll explain how to use long division to find the quotient and remainder for each of these polynomial divisions.
1. (x^3+1) / (x^2+x+2):
Step 1: Start by dividing the highest degree term of the numerator by the highest degree term of the denominator. In this case, x^3 divided by x^2 gives us x.
x
__________
x^2+x+2 | x^3 + 0x^2 + 0x + 1
Step 2: Now we multiply the entire denominator (x^2+x+2) by x and subtract the result from the numerator.
x
__________
x^2+x+2 | x^3 + 0x^2 + 0x + 1
- (x^3 + x^2 + 2x)
-x^2 - 2x + 1
Step 3: Repeat the process by dividing the highest degree term of the new polynomial (-x^2 - 2x + 1) by the highest degree term of the denominator (x^2+x+2). The result is -x.
x - x
__________
x^2+x+2 | x^3 + 0x^2 + 0x + 1
- (x^3 + x^2 + 2x)
---------------
-x^2 - 2x + 1
+ x^2 + x + 2
Step 4: Multiply the entire denominator (x^2+x+2) by -x and subtract the result from the new polynomial (-x^2 - 2x + 1).
x - x
__________
x^2+x+2 | x^3 + 0x^2 + 0x + 1
- (x^3 + x^2 + 2x)
---------------
-x^2 - 2x + 1
+ x^2 + x + 2
---------------
-3x - 1
Step 5: Now we have a new polynomial (-3x - 1). Since the degree of (-3x - 1) is less than the degree of the original denominator (x^2+x+2), we cannot divide it further.
Finally, the quotient is x - x, which simplifies to x, and the remainder is -3x - 1.
2. (14x^3-16x^2+10x+8) / (x-1):
The steps for this example are similar to the previous one. I will skip the detailed explanation and provide the final result.
Quotient: 14x^2 - 2x + 8
Remainder: 0
I hope this explanation helps clarify the long division process for polynomial division!