Given -4i is a root, determine all other roots of f(x) = x^3 - 3x^2 + 16x - 48.
Are they 4i and 3?
complex roots come in conjugate pairs,
so we know two roots to be
-4i and 4i
two factors would be (x-4i) and (x+4i)
or (x^2 + 16) has to be a factor of f(x)
so it would need another linear factor.
since it ends in -48 and our x^2 factor end in +16, the remaining factor must end in -3
by this logic,
f(x) = (x-3)(x^2 + 16)
with roots of 3, 4i, and -4i
check: by Wolfram,
http://www.wolframalpha.com/input/?i=solve+x%5E3+-+3x%5E2+%2B+16x+-+48+%3D+0
To determine the other roots of the polynomial f(x) = x^3 - 3x^2 + 16x - 48, we can use the fact that complex roots occur in conjugate pairs. Since -4i is already a root, we can find the other complex root by taking its conjugate:
Conjugate of -4i is 4i.
Therefore, the other complex root is 4i.
Now let's find the real root(s). We can use polynomial long division or synthetic division to divide f(x) by (x - 3). Performing synthetic division, we get:
3 | 1 -3 16 -48
-----------------------
1 0 16 0
The result of the synthetic division is the quotient: x^2 + 16.
Now, we need to solve the quadratic equation x^2 + 16 = 0 to find the remaining two roots. Rewriting it as (x + 4i)(x - 4i) = 0, we can see that the roots are ±4i.
Therefore, the complete set of roots for f(x) = x^3 - 3x^2 + 16x - 48 is -4i, 4i, and 3.
In conclusion, you are correct. The roots of the polynomial are -4i, 4i, and 3.
To determine the other roots of the equation f(x) = x^3 - 3x^2 + 16x - 48, we can use the fact that complex roots come in conjugate pairs.
Since -4i is a root, its conjugate, 4i, must also be a root. So we have found two of the roots of the equation: -4i and 4i.
To find the third root, we can use the fact that the sum of the roots of a cubic equation is equal to the negation of the coefficient of the quadratic term divided by the leading coefficient.
In this case, the coefficient of the quadratic term is -3, and the leading coefficient is 1. So the sum of the roots is -(-3)/1, which equals 3.
Since we already have two roots (-4i and 4i) with a combined sum of 0, the third root must be -3 in order for the sum of the roots to be 3.
Therefore, the roots of the equation f(x) = x^3 - 3x^2 + 16x - 48 are -4i, 4i, and -3.
In conclusion, your initial hypothesis is incorrect. The roots of the equation are -4i, 4i, and -3, not 4i and 3.