Question

N=3;4and 4i are zero; f(1)=-153

Answer 1

Using the Factor Theorem, the polynomial with zeros 3, 4i and -4i, and f(i) = -153 is given by:

f(x) = 4.5(x³ - 3x² + 16x - 48).

Factor Theorem

The Factor Theorem states that a polynomial function with zeros(also called roots) $x_1, x_2, \codts, x_n$ is given by the following rule, given below.

$f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)$

In which a is the leading coefficient of the polynomial, determining if it is positive(a positive) or negative(a negative).

In the context of this problem, the roots are presented as follows:

$x_1 = 3, x_2 = 4i, x_3 = -4i$

Hence the polynomial is defined as follows:

f(x) = a(x - 3)(x - 4i)(x + 4i)

f(x) = a(x - 3)(x² + 16)

f(x) = a(x³ - 3x² + 16x - 48)

When x = 1, y = -153, hence the leading coefficient is calculated as follows:

-153 = a(1 - 3 + 16 - 48)

-34a = -153

34a = 153

a = 153/34

a = 4.5.

Hence the function is:

f(x) = 4.5(x³ - 3x² + 16x - 48).

More can be learned about the Factor Theorem at brainly.com/question/24380382

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