Question Find the polynomial of minimum degree, with real coefficients, zeros at x=4+4i and x=2, and y-intercept at 64

Answer: Step-by-step explanation: We want to find the minimum-degree polynomial with real coefficients and zeros at: As well as a y-intercept of 64. By the Complex Root Theorem, if a + bi is a root, then a – bi is also a root. So, a third root will be 4 – 4i. The factored form of a polynomial is given by: Where a is the leading coefficient and p and q are the zeros. More factors can be added if necessary. Substitute: Since we want the minimum degree, we won’t need to add any exponents. Expand the second and third factors: Hence: Lastly, we need to determine a. Since the y-intercept is y = 64, this means that when x = 0, y = 64. Thus: Solve for a: Our factored polynomial is: Finally, expand: Reply

Answer:Step-by-step explanation:We want to find the minimum-degree polynomial with real coefficients and zeros at:

As well as a

y-intercept of 64.By the Complex Root Theorem, if

a+bi is a root, thena–bi is also a root.So, a third root will be 4 – 4i.

The factored form of a polynomial is given by:

Where

ais the leading coefficient andpandqare the zeros. More factors can be added if necessary.Substitute:

Since we want the minimum degree, we won’t need to add any exponents.

Expand the second and third factors:

Hence:

Lastly, we need to determine

a. Since they-intercept isy= 64, this means that whenx= 0,y= 64. Thus:Solve for

a:Our factored polynomial is:

Finally, expand: