Question

Can someone help me with this problem asap

Answer 1

Answer:

f(x) = $x^4 - 2x^3 + 49x^2 - 18x + 360$ is the required polynomial.

Step-by-step explanation:

Given the zeroes (roots) of the polynomial are $-3i$ and $2 + 6i$.

We know that complex roots occur in conjugate pairs.

So, this means that $+3i$ and $2 - 6i$ would also be the roots of the polynomial.

If $\pm 3i$ are to be the roots of the polynomial then the polynomial should have been: $x^2 + 9 = 0$.

Now, to determine the polynomial for which $2 \pm 6i$ would be the roots.

Roots of the polynomial are nothing but the values of x (any variable) that would make the polynomial zero.

⇒ $x = 2 + 6i \hspace{35mm} x = 2 - 6i$

⇒ $x - 2 - 6i = 0 \hspace{25mm} x - 2 + 6i = 0$

The required polynomial would be the product of all the above polynomials.

$i.e., (x^2 + 9)(x - 2 + 6i)(x - 2 - 6i) = 0$

Multiply this to get the required equation.

⇒$(x^2 + 9)(x^2 - 2x + 40)$

$\implies x^4 - 2x^3 + 40x^2 + 9x^2 - 18x + 360 = 0$

∴ The required polynomial is x⁴ - 2x³ + 49x² - 18x + 360 = 0.