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1. Let $a,$ $b,$ $c$ be the roots of $x^3 - x + 4 = 0.$ Compute $(a^2 - 1)(b^2 - 1)(c^2 - 1).$

2. The fourth degree polynomial $P(x)$ satisfies $P(1) = 1,$ $P(2) = 2,$ $P(3) = 3,$ $P(4) = 4,$ and $P(5) = 125.$ What is $P(6)?$

3. Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that

$P(1) = P(3) = P(5) = P(7) = a,$and$P(2) = P(4) = P(6) = P(8) = -a.$

What is the smallest possible value of $a$?

Aug 15, 2021

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pls try to have one question per post :)

1.

Let the polynomial be $f(x).$

It is useful to find a polynomial with roots of $$a^2,b^2,c^2$$ so we can quickly solve it using Vieta's.

Notice that $$f(\sqrt{x})$$ has roots of $$a^2, b^2, c^2$$, but only if that root is positive (specifically if the real part is positive). That is because $$\sqrt{x}=k$$ for some constant where the real part is negative has no solutions.

Also, notice that $$f(-\sqrt{x})$$ has roots of $$a^2, b^2, c^2$$, but only of the real part of that root is negative. That means that $$f(\sqrt{x})\cdot f(-\sqrt{x})$$ will ensure that it has roots of $$a^2, b^2, c^2$$, regardless of if the roots are positive or negative.

Also note that since the polynomial $$f(\sqrt{x})\cdot f(-\sqrt{x})$$ is even, it can be expressed as a polynomial in terms of $$\sqrt{x}^2=x$$.

Let g(x) be the polynomial with roots of $$a^2,b^2,c^2$$. We can see that the polynomial, after some simplifying, is equal to:

$$-x^3 + 2 x^2 - x + 16$$

Note that $$(a^2 - 1)(b^2 - 1)(c^2 - 1)=a^2 b^2 c^2 - a^2b^2 - a^2 c^2 - b^2 c^2 + a^2 + b^2 + c^2 - 1$$. By Vieta's, $$a^2b^2c^2=-16, a^2b^2+a^2c^2+b^2c^2=-1, a^2+b^2+c^2=-2$$, and the rest can be solved easily.

2.

This problem can be solved quickly using polynomial interpolation using remainder theorem as follows:

Notice that $$P(x)-x=0$$ for $$x=1, 2, 3, 4$$, which means that $$P(x)-x=A(x-1)(x-2)(x-3)(x-4)$$ for some constant A. Rearranging, we get that $$P(x)=x+A(x-1)(x-2)(x-3)(x-4)$$. We can solve for A by substituting 5 for x:

$$5+A(5-1)(5-2)(5-3)(5-4)=125\\ A(4)(3)(2)=120\\ A=5$$

So the polynomial is equal to $$5+5(x-1)(x-2)(x-3)(x-4)$$.

Substitute x=6 to get the answer.

Aug 16, 2021