third degree with zeros of 2-i,2+i and 2 and a leading coefficient of 4. Construct a polynomial function with stated properties
let's start with the two complex zeros
sum of roots = 2-i + 2+i = 4
product of roots = (2-i)(2+i) = 4 - i^2 = 5
so the quadratic that would form those two zeros is
x^2 - 4x + 5
your function will be
f(x) = 4(x - 2)(x^2 - 4x + 5)
expand and simplify if you have to.
Well, to construct a polynomial function with these properties, we can start by using the zeros you provided. Since the zeros are given as 2-i, 2+i, and 2, we know that our polynomial will have factors (x - (2-i)), (x - (2+i)), and (x - 2).
Firstly, let's multiply the first two factors:
(x - (2-i))(x - (2+i))
We can simplify this by using the difference of squares:
(x - 2 + i)(x - 2 - i)
Expanding further:
(x - 2)(x - 2) - (x - 2)(i) - (x - 2)(-i) + i(-i)
(x - 2)(x - 2) - i(x - 2) + i(x - 2) + 1
Now, let's simplify this expression:
(x^2 - 4x + 4) - (ix - 2i) + (ix - 2i) + 1
x^2 - 4x + 4 - ix + 2i + ix - 2i + 1
x^2 - 4x + 5
So, the polynomial function with a leading coefficient of 4 and zeros 2-i, 2+i, and 2 is:
4(x^2 - 4x + 5)
And there you have it!
To construct a polynomial function with the given properties, we start with the zeros and the leading coefficient.
1. Zeros: Our zeros are 2 - i, 2 + i, and 2.
Since these zeros have complex conjugate pairs (2 - i, 2 + i), we know that they have the form (x - (2 - i))(x - (2 + i)) = ((x - 2) + i)((x - 2) - i).
Expanding this equation gives us:
((x - 2) + i)((x - 2) - i) = (x - 2)² - (i)² = (x - 2)² + 1.
So, our zeros can be expressed as (x - 2)² + 1 = 0.
2. Leading coefficient: The leading coefficient of the polynomial is 4.
Now, combining the zeros and leading coefficient, we can construct the polynomial function:
f(x) = 4((x - 2)² + 1)
Expanding this equation further gives us:
f(x) = 4(x² - 4x + 4 + 1)
f(x) = 4x² - 16x + 20
Therefore, the polynomial function with the stated properties is f(x) = 4x² - 16x + 20.
To construct a polynomial function with the given zeros and leading coefficient, we'll start by finding the factors of the polynomial using the zeros.
The given zeros are 2-i, 2+i, and 2. Since complex zeros occur in conjugate pairs, we can group 2-i and 2+i together.
First, let's find the factors corresponding to each zero. For the zero 2-i, we have (x - (2-i)) as a factor. Simplifying, we get (x - 2 + i). Similarly, for the zero 2+i, we have (x - (2+i)) as a factor, which simplifies to (x - 2 - i). Finally, for the zero 2, we have (x - 2) as a factor.
Now, we can multiply these factors together to get the polynomial function. Since the given polynomial is a third-degree function with a leading coefficient of 4, we can assume the polynomial to be of the form 4(x - 2 + i)(x - 2 - i)(x - 2). Expanding this expression, we have:
4[(x - 2 + i)(x - 2 - i)(x - 2)]
Multiplying these factors out, we get:
4[(x^2 - 4x - xi - 2x + 4 + 2i + xi - 2i + i^2)(x - 2)]
Simplifying further:
4[(x^2 - 6x + 9 - i^2)(x - 2)]
Since i^2 = -1:
4[(x^2 - 6x + 9 + 1)(x - 2)]
4[(x^2 - 6x + 10)(x - 2)]
Now, expanding the remaining terms:
4[x^3 - 6x^2 + 10x - 2x^2 + 12x - 20]
Combining like terms:
4[x^3 - 8x^2 + 22x - 20]
Therefore, the final polynomial function with the given properties is:
f(x) = 4x^3 - 8x^2 + 22x - 20.