Careful with basics

Algebra Level 3

Let $P(x)$ be a polynomial of degree $3$ with real coefficients. Which of the following is possible?

A: $P(x)$ has no real roots.

B: $P(x)$ has exactly $2$ real roots, and they are distinct numbers.

C: $P(1) = -1$ , $P(2) = 1$ , $P(3) = 11$ and $P(4) = 35$ .

D: $i - 2017$ and $i + 2017$ are the roots of $P(x)$ .

Clarification : $i=\sqrt{-1}$ denotes the imaginary unit .

B C D A

1 solution

Anthony Holm
Jun 20, 2017

Any polynomial of degree n with complex coefficients will have exactly n roots (counting multiplicity) in the complex numbers. If it has only real roots then any complex non-real roots will come in conjugate pairs with a+bi and a-bi both being roots. Because of this both A) and B) must be false as P(x) can only have 3 either 0 or 2 non-real roots and thus either 3 or 1 real roots. D) also can't be right as if those 2 numbers were roots, then both -i-2017 and -i+2017 would also be roots leading to 4 roots which is impossible. Finally C) is possible as any polynomial of degree n with real coefficients can be exactly specified with n+1 points that lie on that polynomial. So the four points C) lists exactly specify a degree 3 polynomial with real coefficients just as required.

Careful with basics

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