Form a Polynomial f(x)
Degree: 4, Zeros: 2+2i;-5, Multiplicity:2
f(x)=a(______________)
interpretation:
4 roots:
2+2i
2-2i
-5
-5
the -5 root twice yields (x+5)^2
for the two complex roots:
sum of those two = 4
product of those two = (2+2i)(2-2i) = 8
which gives us the trinomial x^2 -4x + 8
so f(x) = a(x^2 - 4x + 8)(x+5)^2 , where a is any non-zero constant
10
To form the polynomial f(x), we can start by using the given zeros and multiplicities.
1) The first zero is 2+2i. Since complex zeros always come in pairs (if a+bi is a zero, then a-bi is also a zero), we know that the conjugate of 2+2i must also be a zero. The conjugate of 2+2i is 2-2i.
2) The second zero is -5, and it has a multiplicity of 2. This means that it is a repeated zero.
Therefore, the factors of the polynomial are:
(x - (2+2i))(x - (2-2i))(x + 5)(x + 5)
Simplifying further, we have:
(x - 2 - 2i)(x - 2 + 2i)(x + 5)(x + 5)
Expanding the expressions, we get:
(x^2 - 4x + 4)(x^2 + 10x + 25)
Now, we can multiply these two expressions together:
(x^2 - 4x + 4)(x^2 + 10x + 25) = x^4 + 10x^3 + 25x^2 - 4x^3 - 40x^2 - 100x + 4x^2 + 40x + 100
Simplifying further:
x^4 + 6x^3 - 11x^2 + 40x + 100
So the polynomial f(x) is:
f(x) = x^4 + 6x^3 - 11x^2 + 40x + 100
(Note: The coefficient a is not specified in the question, so we did not include it in the final expression for f(x).)
To form a polynomial with the given information, we can start by using the zeros and their multiplicities.
First, let's consider the zero 2+2i. Since complex zeros occur in conjugate pairs, we also have the zero 2-2i. Therefore, the factors associated with this pair of zeros are (x - (2+2i)) and (x - (2-2i)). Simplifying, we get (x - 2 - 2i) and (x - 2 + 2i).
Next, let's consider the zero -5 with multiplicity 2. The factor associated with this zero is (x - (-5))² = (x + 5)².
To find the polynomial f(x), we multiply these factors together:
f(x) = a(x - 2 - 2i)(x - 2 + 2i)(x + 5)²
Now, we can simplify this expression by expanding and multiplying the factors:
f(x) = a[(x - 2)² - (2i)²][(x + 5)(x + 5)]
= a[(x² - 4x + 4) - (-4)][(x + 5)(x + 5)]
= a(x² - 4x + 4 + 4)(x + 5)(x + 5)
= a(x² - 4x + 8)(x + 5)²
Therefore, the polynomial f(x) is given by:
f(x) = a(x² - 4x + 8)(x + 5)²