Find a polynomial function P of the lowest possible degree, having real coefficients, a leading coefficient of 1, and with the given zeros.
2plus2i, minus1, and 2
if r is a root (zero) , then x-r is a factor
complex roots (containing i) occur in conjugate pairs
... if 2+2i is a root , then 2-2i is also
P(x) = (x - 2 - 2i) (x - 2 + 2i) (x + 1) (x - 2)
multiply the factors to find the 4th order function
To find a polynomial function with the given zeros, we can start by using the complex zeros.
Given that one of the zeros is 2 + 2i, we know that its conjugate, 2 - 2i, must also be a zero. This is because complex zeros always come in conjugate pairs.
So, the complex zeros are: 2 + 2i and 2 - 2i.
Next, we have the real zero -1.
To find the polynomial function, we can use the fact that if a number is a zero of a polynomial, then (x - zero) is a factor of the polynomial.
So, the factors corresponding to the zeros are: (x - (2 + 2i)), (x - (2 - 2i)), and (x - (-1)).
Expanding these factors, we get: (x - 2 - 2i), (x - 2 + 2i), and (x + 1).
To find the polynomial function, we multiply these factors together:
P(x) = (x - 2 - 2i)(x - 2 + 2i)(x + 1)
To simplify further, we can use the difference of squares formula for the first two factors:
P(x) = [(x - 2)^2 - (2i)^2](x + 1)
Simplifying the first term, we have:
P(x) = ((x - 2)^2 - 4i^2)(x + 1)
Since i^2 is equal to -1, we can further simplify:
P(x) = ((x - 2)^2 - 4(-1))(x + 1)
P(x) = ((x - 2)^2 + 4)(x + 1)
Expanding the square:
P(x) = (x^2 - 4x + 4 + 4)(x + 1)
P(x) = (x^2 - 4x + 8)(x + 1)
Finally, we can multiply the remaining factors:
P(x) = x^3 + x^2 - 4x^2 - 4x + 8x + 8
Combining like terms, the polynomial function is:
P(x) = x^3 - 3x^2 + 4x + 8
So, the polynomial function with the given zeros is P(x) = x^3 - 3x^2 + 4x + 8.
To find a polynomial function P with the given zeros, we can use the fact that complex zeros come in conjugate pairs. This means that if 2 + 2i is a zero, its conjugate 2 - 2i must also be a zero.
Let's start by writing out the factors of the polynomial. Since 2 + 2i is a zero, the corresponding factor is (x - (2 + 2i)). Simplifying this, we get (x - 2 - 2i). Similarly, for the conjugate 2 - 2i, the factor is (x - (2 - 2i)), which simplifies to (x - 2 + 2i).
Now, since -1 is also a zero, we need another factor: (x - (-1)), which simplifies to (x + 1).
Finally, the polynomial P can be obtained by multiplying all the factors together:
P(x) = (x - 2 - 2i)(x - 2 + 2i)(x + 1)
Now, let's simplify this polynomial:
P(x) = ((x - 2)^2 - (2i)^2)(x + 1)
= ((x - 2)^2 + 4)(x + 1)
= (x^2 - 4x + 4 + 4)(x + 1)
= (x^2 - 4x + 8)(x + 1)
= x^3 - 4x^2 + 8x + x^2 - 4x + 8
= x^3 - 3x^2 + 4x + 8
Therefore, the polynomial function P(x) with the given zeros is:
P(x) = x^3 - 3x^2 + 4x + 8