The quotient is 4x² + 6x + 9 and the remainder is 23.
So, P(2) = 4(2)² + 6(2) + 9 + 23 = 4(4) + 12 + 9 + 23 = 16 + 12 + 9 + 23 = 60.
Hence, P(2) equals 60.
The remainder theorem states that if a polynomial P(x) is divided by (x-a), the remainder is equal to P(a).
To find P(2) for the given polynomial P(x) = -2x² + 4x³ - 3x + 5, we need to divide the polynomial by (x-2) using long division.
Here is the step-by-step process:
1. Write the polynomial in descending order:
P(x) = 4x³ - 2x² - 3x + 5
2. Set up the long division:
2x² - x + 2
---------------------
(x-2) | 4x³ - 2x² - 3x + 5
3. Divide the first term of the polynomial by the first term of the divisor:
4x³ / (x-2) = 4x²
4. Multiply the divisor by the quotient:
(x-2) * 4x² = 4x³ - 8x²
5. Subtract the result from the original polynomial:
(4x³ - 2x² - 3x + 5) - (4x³ - 8x²) = 6x² - 3x + 5
6. Bring down the next term:
6x² - 3x + 5
7. Divide the first term of the new polynomial by the first term of the divisor:
6x² / (x-2) = 6x
8. Multiply the divisor by the new quotient:
(x-2) * 6x = 6x² - 12x
9. Subtract the result from the previous polynomial:
(6x² - 3x + 5) - (6x² - 12x) = 9x + 5
10. Bring down the last term:
9x + 5
11. Divide the first term of the new polynomial by the first term of the divisor:
9x / (x-2) = 9
12. Multiply the divisor by the new quotient:
(x-2) * 9 = 9x - 18
13. Subtract the result from the previous polynomial:
(9x + 5) - (9x - 18) = 23
The remainder is 23.
Therefore, the quotient is 4x² + 6x + 9 and the remainder is 23.
So, P(2) = 4(2)² + 6(2) + 9 + 23 = 4(4) + 12 + 9 + 23 = 16 + 12 + 9 + 23 = 60.
Hence, P(2) equals 60.