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.

-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).

step-by-step explanation: 2

b

step-by-step explanation:

the equation of a line in point- slope form is

y - b = m(x - a)

where m is the slope and (a, b) a point on the line

calculate m using the slope formula

m = (y₂ - y₁ ) / (x₂ - x₁ )

with (x₁, y₁ ) = (0, 4) and (x₂, y₂ ) = (3, - 5) ← 2 points on the line

m = = = - 3

with m = - 3 and (a, b) = (3, - 5), then

y - (- 5) = - 3(x - 3), that is

y + 5 = - 3(x - 3) → b