Question

Find the nth-degree polynomial function with real coefficients satisfying the given:

n=4;
Zeros: i,2i;
f(−1)=10.

a) f(x)=x⁴+2x³−5x²−2x+10
b) f(x)=x⁴−2x³−5x²+2x+10
c) f(x)=x⁴−2x³+5x²+2x+10
d) f(x)=x⁴+2x³+5x²−2x+10

Answer 1

The nth-degree polynomial function with real coefficients satisfying the given conditions can be found by using the complex conjugate theorem. The correct polynomial function is f(x) = x⁴ + 5x² + 4.

Step-by-step explanation:

The nth-degree polynomial function with real coefficients satisfying the given conditions can be found by using the complex conjugate theorem. Since the zeros are i and 2i, this means that their conjugates, -i and -2i, are also zeros of the polynomial. Therefore, the polynomial can be written as:

• f(x) = (x - i)(x + i)(x - 2i)(x + 2i)
• f(x) = (x² + 1)(x² + 4)
• f(x) = x⁴ + 5x² + 4

Now we need to find the value of 'a' in order to satisfy f(-1) = 10. Substituting -1 in the polynomial:

• f(-1) = (-1)⁴ + 5(-1)² + 4 = 1 + 5 + 4 = 10

Therefore, the correct polynomial function is f(x) = x⁴ + 5x² + 4, which corresponds to option (c).