Answer:
(x³ + 3x² - x + 2) ÷ (x - 1) equals x² + 4x - 3 with a remainder of -1.
Explanation:
! :To divide the polynomial (x³ + 3x² - x + 2) by (x - 1) using long division, follow these steps:
Step 1: Write the dividend (x³ + 3x² - x + 2) and divisor (x - 1) in long division format:
___________________
(x - 1) | x³ + 3x² - x + 2
Step 2: Divide the first term of the dividend (x³) by the first term of the divisor (x):
___________________
(x - 1) | x³ + 3x² - x + 2
x²
Step 3: Multiply the divisor (x - 1) by the quotient obtained in step 2 (x²). Write the result below the dividend, aligned with the corresponding term:
___________________
(x - 1) | x³ + 3x² - x + 2
x² + ...
x³ - x²
Step 4: Subtract the result obtained in step 3 (x³ - x²) from the dividend (x³ + 3x² - x + 2):
___________________
(x - 1) | x³ + 3x² - x + 2
x² + ...
_______________
4x² - x
Step 5: Bring down the next term from the dividend, which is -x, and continue the process:
___________________
(x - 1) | x³ + 3x² - x + 2
x² + ... - 4x
4x² - x
_______________
-3x + 2
Step 6: Bring down the last term from the dividend, which is +2, and continue the process:
___________________
(x - 1) | x³ + 3x² - x + 2
x² + ... - 4x - 3
4x² - x
_______________
-3x + 2
-3x + 3
Step 7: Divide the last term of the dividend (-3x + 2) by the first term of the divisor (x):
___________________
(x - 1) | x³ + 3x² - x + 2
x² + ... - 4x - 3
4x² - x
_______________
-3x + 2
-3x + 3
_______
-1
Step 8: Write the remainder (-1) as the last term of the result.
The final result is:
Quotient: x² + 4x - 3
Remainder: -1