Question

Which function has a remainder of 18 when divided by (x−2)?

Select the correct answer below: g(x)=x4+4x2−10x−10
g(x)=2x4+4x2−10x−9 g(x)=x4+4x2−10x−9 g(x)=2x4+x2−9x−10
g(x)=2x4+4x2−

Answer 1

To find the function with a remainder of 18 when divided by (x-2), we substitute x=2 into each option and check if any of them give the desired remainder. None of the given options satisfy this condition.

Step-by-step explanation:

To determine which function has a remainder of 18 when divided by (x-2), we need to use the Remainder Theorem. The Remainder Theorem states that if a polynomial f(x) is divided by (x - a), the remainder is equal to f(a). So, we need to find the function g(x) from the given options that satisfies g(2) = 18.

Let's substitute x = 2 into each option and check which one gives us a remainder of 18:

• For g(x) = x^4 + 4x^2 - 10x - 10, g(2) = 16 + 16 - 20 - 10 = 2. Hence, this option does not give a remainder of 18.
• For g(x) = 2x^4 + 4x^2 - 10x - 9, g(2) = 32 + 16 - 20 - 9 = 19. Hence, this option does not give a remainder of 18.
• For g(x) = x^4 + 4x^2 - 10x - 9, g(2) = 16 + 16 - 20 - 9 = 3. Hence, this option does not give a remainder of 18.
• For g(x) = 2x^4 + x^2 - 9x - 10, g(2) = 32 + 4 - 18 - 10 = 8. Hence, this option does not give a remainder of 18.

None of the given options satisfy g(2) = 18. Therefore, none of the functions have a remainder of 18 when divided by (x-2).

Answer 2

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.