Question

Degree 3 polynomial with integer coefficients with zeros -7i and 45 Select one: a. F (x) = 25x3 - 75x2 + 296x - 112 b. F (x) = 25x3 - 5x2 - 264x + 112 c. F (x) = 5x3 - 4x2 - 245x + 196 d. F (x) = 5x3 - 4x2 + 245x - 196

Answer 1

Answer:

The correct option is (d)

The required polynomial is

F(x)=5x³- 4x²+245x-196

Step-by-step explanation:

Conjugate root theorem:

If the complex number a+ib is a root of a polynomial P(x) with real coefficient in one variable,then the complex conjugate a-ib is also a root of P(x).

Given that,

Degree 3 polynomial with integer coefficients with zeros -7i and $\frac 45$ .

Since -7i is complex root of the required polynomial.

Then 7i is also a root of the required polynomial.

Then,

$F(x) = (x-\frac45)\{x-(-7i)\}(x-7i)$

        $=(x-\frac45)(x+7i)(x-7i)$

        $=(x-\frac45)\{x^2-(7i)^2\}$        [ (a+b)(a-b)= a²-b²]

        $=(x-\frac45)(x^2-49i^2)$

       $=(x-\frac45)(x^2+49)$            [ since i² = -1]

       $=x(x^2+49)-\frac45(x^2+49)$

      $=x^3+49x-\frac45 x^2-\frac 45\times 49$

      =5x³+245x-4x²-196  [ Multiply by 5]

     =5x³- 4x²+245x-196

The required polynomial is

F(x)=5x³- 4x²+245x-196