Find a polynomial of degree 3 with roots 3, 4 - i.

Which one is it?
A.x3 - 10x + 15
B.x3 + 15x2 + 4x - 50
C.x3 - 11x2 + 41x - 51
D.x3 + 24x2 + 3x - 12

Well, 3 is a root of C, and the others are not.

See if Google calculator can do the other root; paste this into the google search engine window:
(4-i)^3-11(4-i)^2 +41(4-i)-51

They all are degree 3.

See if x=3 works for each choice. It does not work for A, B or D.

If one of the choices is correct, it must be (C)

If you want to see if 4-i is a root, long-divide polynomial (C) by (x-3) and see what the roots are for the remaining quadratic, which is
x^2 -8x +17 = 0.
x = (1/2) [8 +/- sqrt(-4)]
= 4 +/- i

3, 4-i then also 4+i

(x-3)(x-4+i)(x-4-i) = 0

(x-3)(x^2 - 8x + 17)

x^3 - 11 x^2 + 41 x - 51

Hello Damon im not understanding... are you saying the answer is c?

We are all saying the answer is C.

X=3, X-3=0. X=4-i, x-4=-i, Square both

sides:X^2-8X+16=-1,X^2-8X+17=0, Multiply:(X-3)(X^2-8X+17)=0
X^3-11X^2+41X-51=0.

Yes, the correct choice is C.

To find a polynomial of degree 3 with the given roots, you need to use the fact that if r is a root of a polynomial, then (x - r) is a factor of that polynomial.

In this case, the given roots are 3 and 4 - i. Since complex roots always come in conjugate pairs, the conjugate of 4 - i is 4 + i. Therefore, the factors of the polynomial are (x - 3), (x - (4 - i)), and (x - (4 + i)).

Next, we'll multiply these factors together to find the polynomial.

(x - 3)(x - (4 - i))(x - (4 + i))

To simplify, we can expand the product, combining like terms.

(x - 3)[(x - 4) + i][(x - 4) - i]

Using the difference of squares, we have:

(x - 3)[(x - 4)^2 - i^2]

Simplifying further:

(x - 3)[(x^2 - 8x + 16) - (-1)]

(x - 3)(x^2 - 8x + 16 + 1)

(x - 3)(x^2 - 8x + 17)

Now, let's multiply the remaining factors to get the final polynomial:

x(x^2 - 8x + 17) - 3(x^2 - 8x + 17)

x^3 - 8x^2 + 17x - 3x^2 + 24x - 51

Combining like terms:

x^3 - 11x^2 + 41x - 51

Therefore, the polynomial of degree 3 with roots 3 and 4 - i is option C) x^3 - 11x^2 + 41x - 51.