What is the result of (x^4+2x^3+x-4)/(x-1)?
Use synthetic division or polynomial long division.
work the problem out:
To perform polynomial long division, we divide the polynomial x^4+2x^3+x-4 by the binomial x-1.
First, set up the division like this:
________
x - 1 | x^4 + 2x^3 + x - 4
Now, we start dividing. We divide the leading term of the polynomial (x^4) by the leading term of the binomial (x), giving us x^3. We write this above the division line.
x^3
__________
x - 1 | x^4 + 2x^3 + x - 4
Next, we multiply the binomial (x-1) by the result we obtained (x^3). In this case, (x^3)(x) = x^4. We write this below the polynomial, aligning it with the like term.
x^3
__________
x - 1 | x^4 + 2x^3 + x - 4
- x^4
Now, we subtract. We bring down the next term of the polynomial (2x^3). And since we have a negative x^4, we need to change the signs of all the terms in this step. So, we subtract 2x^3 from x^4, which gives us -x^4. We write this below the subtraction line.
x^3
__________
x - 1 | x^4 + 2x^3 + x - 4
- x^4
__________
x^3
We repeat the process. We divide the leading term of the new polynomial (x^3) by the leading term of the binomial (x), giving us x^2. We write this above the division line.
x^3
__________
x - 1 | x^4 + 2x^3 + x - 4
- x^4
__________
x^3
- x^3
Next, we multiply the binomial (x-1) by the result we obtained (x^2). In this case, (x^2)(x) = x^3. We write this below the polynomial, aligning it with the like term.
x^3
__________
x - 1 | x^4 + 2x^3 + x - 4
- x^4
__________
x^3
- x^3
__________
0
Since the degree of the dividend (x^4+2x^3+x-4) is 4 and the degree of the divisor (x-1) is 1, we have reached a remainder of 0, indicating that the division is complete.
Therefore, the result of (x^4+2x^3+x-4)/(x-1) is x^3+x^2+0, which simplifies to x^3+x^2.