Final answer:
To solve the expression (x⁶-2x⁵+x⁴-x³+3x²-x+24)/(x+2) using long division, divide the terms of the numerator by the denominator, and subtract the results from the numerator until the remainder is smaller than the denominator.
Step-by-step explanation:
To solve the expression using long division:
- Divide the first term of the numerator by the denominator.
- Multiply the quotient by the denominator, and subtract it from the numerator.
- Repeat the process until the numerator is smaller than the denominator.
- The final quotient is the answer.
In this case, the expression is (x⁶-2x⁵+x⁴-x³+3x²-x+24)/(x+2).
Following the steps of long division, we have:
x⁵-3x³+7x²-13x+26
_____________
x+2 ∣ x⁶-2x⁵+x⁴-x³+3x²-x+24
-x⁶-2x⁵
_____________
0x⁵+x⁴-x³+3x²
0x⁵+2x⁴
_____________
-2x⁴-x³+3x²-x
-2x⁴-4x³
_____________
3x³+3x²-x
3x³+6x²
_____________
-9x²-x
-9x²-18x
_____________
17x+24
17x+34
_____________
-10
So, the solution to the expression (x⁶-2x⁵+x⁴-x³+3x²-x+24)/(x+2) using long division is x⁵-3x³+7x²-13x+26 with a remainder of -10.