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julia-pushkina [17]
3 years ago
8

Which of the following is a polynomial with roots 2, 3i, and −3i? f(x) = x3 − 2x2 + 6x − 9 f(x) = x3 − 6x2 + 9x − 18 f(x) = x3 −

6x2 + 18x − 2 f(x) = x3 − 2x2 + 9x − 18
Mathematics
2 answers:
Andreas93 [3]3 years ago
4 0
F(x) = x3 − 2x2 + 9x − 18
vagabundo [1.1K]3 years ago
3 0

Answer:

f(x)=x^3-2 x^2+9 x-18

Step-by-step explanation:

The roots of the polynomial are 2,3i,-3i.

This implies that x-2,x-3i,x+3i are factors of the given polynomial.

The polynomial will have equation;

f(x)=(x-2)(x-3i)(x+3i)

We expand using difference of two squares on the complex conjugates to get;

f(x)=(x-2)(x^2-(3i)^2)

\Rightarrow f(x)=(x-2)(x^2-(-3)^2(i)^2).

\Rightarrow f(x)=(x-2)(x^2-9(i)^2).

Recall that;

\boxed{i^2=-1}

\Rightarrow f(x)=(x-2)(x^2+9).

Expand using the distributive property to get;

\Rightarrow f(x)=x^3+9x-2x^2-18.

We rewrite in standard form to obtain;

f(x)=x^3-2x^2+9 x-18

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