Final answer:
The polynomial with a remainder of 18 when divided by (x-2) is g(x) = 2x⁴ + 4x² − 10x − 9, found by applying the Remainder Theorem.
So, the correct answer is option 2) g(x) = 2x⁴ + 4x² − 10x − 9.
Step-by-step explanation:
The function which has a remainder of 18 when divided by (x−2) can be found using the Remainder Theorem. The Remainder Theorem states that if a polynomial f(x) is divided by (x−c), the remainder of the division is f(c). To find the polynomial that leaves a remainder of 18 when divided by (x−2), we plug x = 2 into each of the given polynomial options and look for a value of 18.
Let's apply the Remainder Theorem to each option:
- g(x) = x⁴ + 4x² − 10x − 10: g(2) = 2⁴ + 4(2)² − 10(2) − 10 = 16 + 16 − 20 − 10 = 2
- g(x) = 2x⁴ + 4x² − 10x − 9: g(2) = 2(2)⁴ + 4(2)² − 10(2) − 9 = 32 + 16 − 20 − 9 = 18
- g(x) = x⁴ + 4x² − 10x − 9: g(2) = 2⁴ + 4(2)² − 10(2) − 9 = 16 + 16 − 20 − 9 = 3
- g(x) = 2x⁴ + x² − 9x − 10: g(2) = 2(2)⁴ + (2)² − 9(2) − 10 = 32 + 4 − 18 − 10 = 8
- g(x) = 2x⁴ + 4x² − 10: g(2) = 2(2)⁴ + 4(2)² − 10 = 32 + 16 − 10 = 38
Thus, the correct polynomial is 2x⁴ + 4x² − 10x − 9, which yields the remainder of 18 when divided by (x−2).
So, the correct answer is option 2) g(x) = 2x⁴ + 4x² − 10x − 9.