When using a counterexample to prove a conditional statement false, which must be true about the counterexample?
2 answers:
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A polynomial of degree 3 has exactly 3 solutions if we use complex numbers. If the polynomial has real coefficients and z is a complex solution, i.e., , the conjugate is also a solution:
So, in the first case, if we know that 1-i is a solution, the conjugate 1+i must be a solution as well.
So, we know all the roots: -5, 1+i, 1-i.
When you know the solutions of a polynomial, you can write the polynomial (up to multiples) as
So, in your case, we have
You can check that indeed , because the constant term is 10, so, we're ok with this
The second exercise is exactly the same: the solutions -6, i, -i identify the polynomial
In this case, P(-3)=30, so if we want P(-3)=60 we have to multiply everything by 2: