Question

Using the remainder theorem, which binomial divisor of f(x) results in a remainder of -90,916 for the function: f(x)=x5−4x3+51x2−16? a: (x + 1) b: (x - 2) c: (x - 3) d: (x + 4)

Answer 1

The binomial divisor of f(x) that results in a remainder of -90,916 is (x - 3).

Step-by-step explanation:

To find the binomial divisor using the remainder theorem, we need to substitute each option into the function f(x) = x5 -
`4x^3 + 51x^2 - 16` and check for a remainder of -90,916. Let's do the calculations for each option:

1. (x + 1): When we divide f(x) by (x + 1), the remainder is -109, which is not -90,916.
2. (x - 2): When we divide f(x) by (x - 2), the remainder is -114, which is not -90,916.
3. (x - 3): When we divide f(x) by (x - 3), the remainder is -90,916, which matches the desired value.
4. (x + 4): When we divide f(x) by (x + 4), the remainder is 284, which is not -90,916.

Therefore, the correct binomial divisor is (x - 3) (option c).

Answer 2

The binomial divisor that results in a remainder of -90,916 for the function f(x) = x^5 - 4x^3 + 51x^2 - 16 is (x + 4).

Step-by-step explanation:

To find which binomial divisor of f(x) results in a remainder of -90,916 for the function f(x) = x^5 - 4x^3 + 51x^2 - 16, we can use the remainder theorem. The remainder theorem states that if you divide a polynomial f(x) by a binomial divisor x - c, the remainder is equal to f(c). So, we need to check which of the given binomial divisors gives us a remainder of -90,916 when substituted into the function.

We can substitute each binomial divisor into the function:

a: (x + 1) -> f(1) = (1)^5 - 4(1)^3 + 51(1)^2 - 16 = -34

b: (x - 2) -> f(2) = (2)^5 - 4(2)^3 + 51(2)^2 - 16 = 4

c: (x - 3) -> f(3) = (3)^5 - 4(3)^3 + 51(3)^2 - 16 = 326

d: (x + 4) -> f(-4) = (-4)^5 - 4(-4)^3 + 51(-4)^2 - 16 = -90,916

Therefore, the binomial divisor that results in a remainder of -90,916 is (x + 4), option d.