Question

Construct a polynomial function with the following properties: third degree, only real coefficients, −1 and 3+i are two of the zeros, y-intercept is −40.

Answer 1

Answer:

-4x^3 + 20x^2 - 16x - 40.

Step-by-step explanation:

Complex zeroes exist in conjugate pairs so the third root must be 3 - i.

So knowing the 3 zeroes we can write:

(x + 1)(x - (3 - i))(x - (3 + i)) = 0

= (x + 1)(x - 3 + i)(x - 3 - i)

= (x + 1)( x^2 - 3x - ix - 3x + 9 + 3i  + ix - 3i - i^2)

= (x + 1)(x^2 - 6x + 10)

= x^3 - 6x^2 + 10x + x^2 - 6x + 10

= x^3 - 5x^2 + 4x + 10.

Now the y -intercept is -40 so substituting  x = 0 in the polynomial

y = 10

We have multiply the whole equation by -4). giving

-4x^3 + 20x^2 - 16x - 40.  (answer).