The correct answer is:
x³+x²+x+1.
Explanation:
First we write the polynomial out with all of the other powers of x, using 0 as their coefficients:
(1x⁴+0x³+0x²+0x-1)÷(x-1)
To perform synthetic division, we write the coefficients of the dividend in a row:
1 0 0 0 1
We take the 1 from x-1 and use it in the box. We then drop the first 1 from the row of coefficients down.
We multiply our 1 in the box by the 1 at the bottom; this is 1 and goes under the first 0 to the right of the 1 in the coefficients row. We now add this 0+1; this is 1 and goes at the bottom beside the other 1.
Multiply this by 1; this is 1 and goes under the second 0 from the left in the row of coefficients. Add this to the 0; this is 1 and goes at the bottom, to the right of the other two 1's.
Multiply this by 1; this is 1 and goes under the third and last 0 in the row of coefficients. Add this to the 0; this is 1 and goes at the bottom, to the right of the other three 1's.
Multiply this by 1; this is 1 and goes under the -1 in the row of coefficients. Add this to the -1; this is 0 and goes at the bottom, to the right of the four 1's. This 0 means there is no remainder and the quotient was evenly divided. It gives us
1 1 1 1 0
This means we have 1x³+1x²+1x+1 with no remainder.
