Question

f(x)=x*4-2x*3-50x*2+366x-675

Answer 1

Given the equation;

$f(x)=x^4-2x^3-5x^2+366x-675$

And one of the roots of the equation is;

$4-3i$

According to Conjugate Roots Theorem, If a polynomial has a root a+bi where a and b are real numbers, it must also have a root a-bi.

Apply the root to the given case, since the given polynomial has a root 4-3i, the it must also have a root 4+3i.

So, we have two of the roots of the polynomial;

$\begin{gathered} 4-3i \\ \text{and} \\ 4+3i \end{gathered}$

Using the roots, let us derive a factor polynomial of the given polynomial;

$\begin{gathered} \mleft(x-\mleft(4-3i\mright)\mright)\mleft(x-\mleft(4+3i\mright)\mright) \\ =\mleft(x-4+3i\mright)\mleft(x-4-3i\mright) \\ \text{Expanding;} \\ =\mleft(x^2-4x-3ix-4x+16+12ix+3ix-12ix-9i^2\mright) \\ \text{recall that i}^2=-1\text{ and collecting the like terms} \\ =\mleft(x^2-4x-3ix-4x+16+12ix+3ix-12ix-9(-1)^{}\mright) \\ =(x^2-4x-4x+12ix-12ix-3ix+3ix+16-9(-1)^{}) \\ =(x^2-4x-4x+12ix-12ix-3ix+3ix+16+9^{}) \\ =(x^2-8x+25^{}) \end{gathered}$

So, the derived equation is a factor of the the given polynomial.

To get the other factors we will divide the given polynomial by the derived polynomial above;

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