Alright, let's go through the division process. We are dividing the polynomial 2x³-7x²+10x-8 by x-2
Step 1:
Divide the leading term of the dividend (2x³) by the leading term of the divisor (x). This yields the first term of the quotient: 2x².
Step 2:
Multiply the divisor (x-2) by the first term of the quotient (2x²), then subtract this from the original dividend. This gives us a new polynomial: -3x²+10x-8.
Step 3:
Now, we will repeat process with the new polynomial. By dividing the leading term of the new polynomial (-3x²) by the leading term of the divisor (x), we get the second term of the quotient: -3x.
Step 4:
Multiply the divisor (x-2) by the new term of the quotient (-3x), then subtract this result from the leftover polynomial from the previous step. This gives us a new polynomial: 4x-8.
Step 5:
Once again, let's repeat the process with the new polynomial. Divide the leading term of the new polynomial (4x) by the leading term of the divisor (x) to get the third term of the quotient: 4.
Step 6:
Multiply the divisor (x-2) by the new term of the quotient (4), then subtract it from the leftover polynomial: 0. There are no more terms left in the polynomial, which means we are done, and this is the remainder.
So, the polynomial 2x³-7x²+10x-8 divided by x-2 gives a quotient of 2x²-3x+4 and a remainder of 0. This means that the polynomial 2x³-7x²+10x-8 is exactly divisible by x-2.