A polynomial f(x) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f(x) as a product of linear and/or quadratic polynomials with real coefficients that are irreducible over .
3, −3 − 2i; degree 3
a zero of 3 gives us a factor of (x-3)
complex zeros always come in conjugate pairs, so if
-3-2i is one of them, -3+2i must be another.
factors are (x - (-3-2i)) and (x - (-3+2i))
or (x + 3 + 2i) and (x + 3 - 2i)
multiply the last two to get a quadratic factor (......)
your f(x) = (x-3)(......)
To find the polynomial, we start by using the given zeros to form the factors.
1. The zero 3 corresponds to the linear factor (x - 3).
2. The zero -3 - 2i corresponds to the complex conjugate (−3 + 2i).
Now we have two factors: (x - 3) and (x - (-3 + 2i)).
To express the complex conjugate as a quadratic polynomial, we need to multiply it with its conjugate.
3. We multiply (−3 + 2i) with its conjugate (-3 - 2i) to get a quadratic factor: (x - (-3 + 2i))(x - (-3 - 2i)) = ((x + 3 - 2i)(x + 3 + 2i))
This simplifies to: (x + 3)^2 - (2i)^2 = (x + 3)^2 - 4i^2 = (x + 3)^2 + 4.
Now we have the factors: (x - 3) and ((x + 3)^2 + 4).
To calculate the degree of the polynomial, we sum up the degrees of all the factors.
- The degree of (x - 3) is 1.
- The degree of ((x + 3)^2 + 4) is 2.
The total degree is 1 + 2 = 3, which matches the given degree.
Finally, we can express the polynomial f(x) as a product of irreducible factors:
f(x) = (x - 3)((x + 3)^2 + 4)
To express the polynomial f(x) as a product of irreducible polynomials, we first need to determine its factors by using the given zeros.
The zeros of the polynomial are 3, -3, and -2i. Since the polynomial has real coefficients, the complex conjugate of -2i is also a zero, which is 2i.
Now we can write the factors of f(x) as follows:
(x - 3)(x + 3)(x + 2i)(x - 2i)
To simplify further, we can multiply the complex conjugate factors:
(x - 3)(x + 3)(x^2 + 4)
Thus, the polynomial f(x) with the given zeros and degree can be expressed as the product of the irreducible polynomials (x - 3)(x + 3)(x^2 + 4).