Write a polynomial function in standard form that has the given zeros and has a leading coefficient of 1 for: -2, 4+i
Write a polynomial function of least degree that has the given zeros.2 -3 4 -6
To write a polynomial function with given zeros, we start by using the zero-product property. This property states that if a polynomial function has a zero at a particular value, then its corresponding factor is equal to zero.
First, let's consider the zero -2. To build a factor that corresponds to this zero, we subtract -2 from a variable, let's say x. The resulting expression will be x + 2.
Next, let's consider the zero 4 + i, where i is the imaginary unit. Since complex zeros occur in conjugate pairs, the conjugate of 4 + i is 4 - i. So, our factors corresponding to the zeros 4 + i and 4 - i will be (x - (4 + i)) and (x - (4 - i)) respectively. Simplifying these expressions gives us (x - 4 - i) and (x - 4 + i).
Finally, to construct the polynomial function, we multiply all the factors together:
(x + 2) * (x - 4 - i) * (x - 4 + i)
To simplify this expression further, we can multiply the factors by their conjugates to get rid of the imaginary terms:
(x + 2) * ((x - 4)^2 - (i)^2)
Simplifying even more gives us:
(x + 2) * ((x - 4)^2 + 1)
Expanding this expression, we get:
(x + 2) * (x^2 - 8x + 16 + 1)
Multiplying again:
(x + 2) * (x^2 - 8x + 17)
Now, let's multiply these two terms:
x^3 - 8x^2 + 17x + 2x^2 - 16x + 34
Simplifying further:
x^3 - 6x^2 + x + 34
Therefore, the polynomial function in standard form with the given zeros (-2, 4+i) and a leading coefficient of 1 is:
f(x) = x^3 - 6x^2 + x + 34.