Find a polynomial function with integer coefficients with degree 4, one root 3+radical2, and another root 4-3i.
I got x^4-14x^3+80x^2-206x+175
I'm I correct?
correct. good job
To determine whether your polynomial function is correct, we can apply the fact that if a complex number is a root of a polynomial with real coefficients, then its conjugate is also a root.
Given that one root is 3 + √2, we know that its conjugate is 3 - √2 because the coefficients of the polynomial function must have integer values. Similarly, if another root is 4 - 3i, its conjugate would be 4 + 3i.
To find a polynomial with these roots, we can set up the factors:
(x - (3 + √2))(x - (3 - √2))(x - (4 - 3i))(x - (4 + 3i))
Expanding this expression, we get:
(x - 3 - √2)(x - 3 + √2)(x - 4 + 3i)(x - 4 - 3i)
Multiplying the first two factors using the difference of squares formula:
[(x - 3)^2 - (√2)^2][(x - 4 + 3i)(x - 4 - 3i)]
Simplifying further:
[(x^2 - 6x + 9) - 2][(x - 4)^2 - (3i)^2]
With more simplification:
(x^2 - 6x + 7)(x^2 - 8x + 16 + 9)
Expanding this expression:
(x^2 - 6x + 7)(x^2 - 8x + 25)
Multiplying the two binomials:
x^4 - 8x^3 + 25x^2 - 6x^3 + 48x^2 - 150x + 7x^2 - 56x + 175
Combining like terms:
x^4 - 14x^3 + 80x^2 - 199x + 175
So, your polynomial function is:
f(x) = x^4 - 14x^3 + 80x^2 - 199x + 175.
Therefore, you are correct!