Could you please check my answers?
Find an nth degree polynomial function with real coefficients satisfying the given conditions.
1. n=3; 3 and i are zeros; f(2)=20
-I got: f(x)=-4^3+12x^2-4x+12
3.n=3;4 and i zeros;f(-3)=60
-I got:f(x)=6x^3+24x^2+6x+24
Could you please check my answers?
Find an nth degree polynomial function with real coefficients satisfying the given conditions.
1. n=3; 3 and i are zeros; f(2)=20
-I got: f(x)=-4^3+12x^2-4x+12
3.n=3;4 and i zeros;f(-3)=60
-I got:f(x)=6x^3+24x^2+6x+24
To find an nth degree polynomial function with real coefficients satisfying the given conditions, we can use the fact that if a polynomial has a complex root, its conjugate will also be a root.
For the first problem, n=3 and the zeros are 3 and i. Since i is a complex number, its conjugate is -i, which means -i is also a zero. Now we have all three zeros: 3, i, and -i.
To find the polynomial, we can use the fact that if we know the zeros, we can express the polynomial as the product of its linear factors. In this case, the linear factors are (x-3), (x-i), and (x+i).
To find the polynomial with real coefficients, we need to multiply the complex conjugate factors together. So the polynomial becomes (x-3)(x-i)(x+i).
Expanding this expression, we get:
(x-3)(x-i)(x+i) = (x-3)(x^2 - i^2)
= (x-3)(x^2 + 1)
= x^3 + 1x^2 - 3x^2 - 3x
Simplifying further:
= x^3 - 2x^2 - 3x
Now we can substitute f(2) = 20 into the polynomial and solve for the coefficient:
f(2) = 20
2^3 - 2(2)^2 - 3(2) = 20
8 - 2(4) - 6 = 20
8 - 8 - 6 = 20
-6 = 20
Since the equation is not satisfied, it seems that there might be an error in your calculation. Please recheck your steps.
For the second problem, n=3 and the zeros are 4 and i. Again, since i is a complex number, its conjugate is -i, which means -i is also a zero. Now we have all three zeros: 4, i, and -i.
Using the same approach, we can express the polynomial as (x-4)(x-i)(x+i).
Expanding this expression, we get:
(x-4)(x-i)(x+i) = (x-4)(x^2 - i^2)
= (x-4)(x^2 + 1)
= x^3 + 1x^2 - 4x^2 - 4x
Simplifying further:
= x^3 - 3x^2 - 4x
Now we can substitute f(-3) = 60 into the polynomial and solve for the coefficient:
f(-3) = 60
(-3)^3 - 3(-3)^2 - 4(-3) = 60
-27 - 3(9) +12 = 60
-27 - 27 +12 = 60
-54 + 12 = 60
Since the equation is not satisfied, it appears that there might be a mistake in your calculation. Please double-check your steps.