How to write a polynomial function with the integral coefficients that have the following roots:
#1.) 0, -1/2, 6
#2.) + or - 5i
#1 x(x+ 1/2)(x-6) = 0
Multiply it out to get the (cubic) polynomial.
#2 (x- 5i)(x +5i) = 0
Multiply it out and remember that i^2 = -1
1,2,3
To write a polynomial function with the given roots, we will use the fact that the roots of a polynomial equation correspond to the solutions of the equation when it is set to zero. Here's how you can write the polynomial function for each case:
#1.) Roots: 0, -1/2, 6
To write a polynomial with these roots, we use the fact that if a number "r" is a root of a polynomial, then "x - r" is a factor of the polynomial. So for the given roots, the factors are (x - 0), (x - (-1/2)), and (x - 6). We can multiply these factors to get the polynomial:
(x - 0)(x - (-1/2))(x - 6)
Simplifying:
x(x + 1/2)(x - 6)
So the polynomial function with those roots is:
f(x) = x(x + 1/2)(x - 6)
#2.) Roots: ±5i
Since the roots are complex (±5i), the conjugates of these roots will also be roots of the polynomial equations. Therefore, we have the roots (5i) and (-5i).
Complex roots always come in conjugate pairs.
To create a polynomial with these roots, we again use the fact that if "r" is a root, then "x - r" is a factor. So the factors for these roots are (x - 5i) and (x + 5i). Multiplying these factors will give us the polynomial:
(x - 5i)(x + 5i)
Using the difference of squares to simplify:
(x^2 - (5i)^2)
(x^2 - 25(i^2))
Note that i^2 = -1 (since i is defined as the square root of -1),
(x^2 - 25(-1))
(x^2 + 25)
Therefore, the polynomial function with those roots is:
f(x) = x^2 + 25
Remember that the coefficients of the polynomial are the integral coefficients, since all the roots were given as integers or complex numbers.