Question

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Answer 1

Simplest polynomial function is $x^4-x^3-11x^2+9x+18.$

Solution:

Given data:

Zeroes are 3i, –1, 2.

3i is a complex root of the function.

If 3i is a zero of the polynomial then –3i is also a zero of the polynomial.

Therefore zeroes are 3i, –3i –1, 2.

By factor theorem,

If a is zero of the function, then (x – a) is a factor of the polynomial.

So, the factors are (x – 3i), (x + 3i), (x + 1), (x –2).

On multiplying the factors, we get the polynomial.

$(x - 3i) (x + 3i) (x + 1) (x -2)$

            $=(x^2+3ix-3ix+(3i)^2)(x^2-2x+x-2)$

            $=(x^2-9)(x^2-x-2)$

Since the value of $i^2=-1$

            $=x^2(x^2-x-2)-9(x^2-x-2)$

            $=x^4-x^3-2x^2-9x^2+9x+18$

            $=x^4-x^3-11x^2+9x+18$

Simplest polynomial function is $x^4-x^3-11x^2+9x+18.$