Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros.
3-i,sqrt2
f(x)=??
since the coefficients are rational, the roots come in conjugate pairs. So, the roots are
3-i, 3+i, √2, -√2
f(x) = (x-(3-i))(x-(3+i))(x-√2)(x+√2)
= (x^2-6x+9+1)(x^2-2)
= ...
To find a polynomial function with rational coefficients that has the given numbers as some of its zeros, we can use the fact that complex conjugate pairs are zeros of polynomial functions with real coefficients.
In this case, we have the complex number 3 - i as one of the zeros. The complex conjugate of 3 - i is 3 + i, which is also a zero. Additionally, we have sqrt(2) as another zero.
To find the polynomial function, we need to multiply the factors corresponding to each zero. Let's start with the complex conjugate zeros: (x - (3 - i))(x - (3 + i)).
Expanding this expression gives us:
(x - 3 + i)(x - 3 - i) = ((x - 3) + i)((x - 3) - i)
= (x - 3)^2 - i^2
= (x - 3)^2 - (-1)
= (x - 3)^2 + 1
Now let's include the sqrt(2) as another zero. We multiply the expression above by (x - sqrt(2)):
(x - sqrt(2))((x - 3)^2 + 1)
Expanding this expression gives us:
(x - sqrt(2))(x - 3)^2 + (x - sqrt(2))
= (x - sqrt(2))(x^2 - 6x + 9) + (x - sqrt(2))
Multiplying these expressions, we get:
x^3 - 6x^2 + 9x - x*sqrt(2) - sqrt(2)*x^2 + 6x - 9 - sqrt(2)
Simplifying and regrouping the terms gives us:
x^3 - (7 + sqrt(2))x^2 + (15 - sqrt(2))x - (9 + sqrt(2))
Thus, the polynomial function with the given zeros and rational coefficients is:
f(x) = x^3 - (7 + sqrt(2))x^2 + (15 - sqrt(2))x - (9 + sqrt(2))
To find a polynomial function with rational coefficients that has 3 - i and √2 as zeros, we can use the fact that complex zeros occur in conjugate pairs. This means that if 3 - i is a zero, then its conjugate 3 + i is also a zero.
Therefore, the factors of the polynomial are (x - (3 - i)) and (x - (3 + i)).
Simplifying these factors, we get:
(x - 3 + i) and (x - 3 - i)
Now, to include √2 as a zero, we need another factor of (x - √2).
Multiplying all these factors together, we get the polynomial:
f(x) = (x - 3 + i)(x - 3 - i)(x - √2)
Expanding this polynomial further, we get:
f(x) = (x^2 - 6x + 10)(x - √2)
Multiplying these expressions together, we get:
f(x) = x^3 - (√2 + 6)x^2 + (10√2 - 6√2)x - 10√2