Can you find the fourth degree polynomial with integer coefficients that has given roots for 2i and 4-i
if R is a root (zero), then x-R is a factor
complex roots (with i's in them) occur in conjugate pairs
so the roots are ... 2i , -2i , 4-i , 4+i
the factors are ... (x - 2i) (x + 2i) (x + 4 - i) (x + 4 + i)
multiply the factors together to find the polynomial
if a+bi is a root, then so is a-bi
(x-(a+bi))(x-(a-bi)) = (x-a)^2 + b^2
So, your polynomial will be
(x^2+4)((x-4)^2+1) = (x^2+4)(x^2-8x+17) = ...
oops...got the signs wrong in the last two factors
... can't follow my own instructions
should be ... (x - 2i) (x + 2i) (x - 4 - i) (x - 4 + i)
Yes, I can help you find the fourth degree polynomial with integer coefficients that has the given roots.
To find the polynomial, we can use the fact that complex roots always come in conjugate pairs. This means that if 2i is a root, then -2i must also be a root.
Now we can set up the factors for the polynomial using the roots:
(x - 2i)(x + 2i)(x - (4 - i))(x - (4 + i))
To simplify this expression, we can expand it using the FOIL method:
[(x^2 - (2i)^2)(x - (4 - i))(x - (4 + i)]
Simplifying further:
[(x^2 + 4)(x - 4 + i)](x - 4 - i)
Now we can continue expanding:
[(x^2 + 4)(x^2 - 4x + xi - 4x + 16 - 4i - xi + 4i + 1)]
Simplifying again:
[(x^2 + 4)(x^2 - 8x + 17)]
Finally, we can multiply the remaining factors:
x^4 - 8x^3 + 17x^2 + 4x^2 - 32x + 68
Combining like terms:
x^4 - 8x^3 + 21x^2 - 32x + 68
Thus, the fourth degree polynomial with integer coefficients that has the given roots is:
x^4 - 8x^3 + 21x^2 - 32x + 68