Answer:
x²-x+1
Step-by-step explanation:
We write the dividend, x⁴+x³+7x²-6x+8, under the division box. We write the divisor, x²+2x+8, outside the box to the left.
We see how many times x² goes into x⁴. It will go x² times. We write this on the line above the dividend. We multiply back through;
x²(x²+2x+8) = x⁴+2x³+8x²
We write this under the first three terms of the dividend. We then subtract:
(x⁴+x³+7x²)-(x⁴+2x³+8x²) = -1x³-1x²
This goes below the subtraction. We bring down the next term, -6x.
Next we see how many times x² goes into -1x³. It goes -x times; we write this beside the x² in the quotient. We multiply back through:
-x(x²+2x+8) = -x³-2x²-8x
This goes below the -1x³-1x²-6x. We subtract the two:
(-1x³-1x²-6x)-(-x³-2x²-8x) = -1x³+1x³-1x²+2x²-6x+8x = 1x²+2x
We write this below the last line in the work. We bring down the last term, 8.
Now we see how many times x² goes into 1x². It goes 1 time; since this is positive, we write +1 beside the answer we have at the top. Multiply back through:
1(x²+2x+8) = x²+2x+8
Now we subtract:
(x²+2x+8)-(x²+2x+8) = 0
This makes our answer x²-x+1
