Question

Write an equation for the polynomial function that has 2 and -3 + 5i as zero

Answer 1

Answer:

$p(x) = {x}^{3} + 4{x}^{2} + 22x- 68$

Step-by-step explanation:

We want to write an equation for the polynomial function with roots;

$x = 2$

and

$x = - 3 + 5i$

Note that complex roots come in pairs and in conjugates.

Therefore

$x = - 3 - 5i$

is also a root.

By the factor theorem:

(x-2), (x+3+5i), and (x+3-5i) are factors of the polynomial.

Let p(x) be the polynomial, then in factored form.

$p(x) = (x - 2)(x + 3 + 5i)(x + 3 - 5i)$

We expand now to get:

$p(x) = (x - 2)( {x}^{2} + 3x - 5ix + 3x + 9 - 15i + 5ix + 15i + 25)$

Simplify now

$p(x) = (x - 2)( {x}^{2} + 6x +34)$

We expand further;

$p(x) = x( {x}^{2} + 6x + 34) - 2( {x}^{2} + 6x + 34)$

$p(x) = {x}^{3} + 6 {x}^{2} + 34x - 2 {x}^{2} - 12x - 68$

$p(x) = {x}^{3} + 4{x}^{2} + 22x- 68$