Write an equation of a polynomial of degree 4 and integer coefficients with roots -2i, 1/3 and 5.
Hello,
If a is a root of a polynomial, then x-a divides this polynomial.
So you can say that (x-(-2i))(x-1/3)(x-5) divides the polynomial.
What about the 4th factor ? Since the coefficients are integers, if a complex number (like -2i) is a root, then its conjugate is also a root.
The conjugate of a complex number a+ib is a-ib. Here, a=0 and b=-2. Can you do it ?
1. I assume that this polynomial has +2i as a root too.
2. $\displaystyle p(x)=a\left((x^2+4)(x-5)\left(x-\frac13\right)\right)$
3. Expand the brackets:
$\displaystyle p(x)=a\left(x^4-\frac{16}3 x^3 + \frac{17}3 x^2 - \frac{64}3 x + \frac{20}3 \right)$
4. Choose a = 3 and you'll get:
$\displaystyle p(x)=3x^4-16x^3+17x^2-64x+20$
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EDIT: Too fast for me, Moo