User:Acfmathteam

Problems
Sum of Coefficients

1) Find the sum of the coefficients of the expanded form of $$P(x)=(3x^2-4x+2)^5$$.

2) (ARML 1989) If $$P(x)$$ is a polynomial in $$x$$, and $$x^{23} + 23x^{17} - 18x^{16} - 24x^{15} + 108x^{14} = (x^4 - 3x^2 - 2x + 9)$$· $$P(x)$$ for all values of $$x$$, compute the sum of the coefficients of $$P(x)$$.

Vieta's Formulae

3) Determine the product of the roots of $$50x^{50} + 49x^{49} + ... + x + 1 = 0$$.

4) Find all ordered pairs $$(x, y, z)$$ that satisfy:

$$x + y + z = 17$$

$$xy + yz + xz = 94$$

$$xyz = 168$$

5) Find all ordered triples $$(x, y, z)$$ that satisfy:

$$x + y - z = 0$$

$$zx - xy + yz = 27$$

$$xyz = 54$$

5) (AIME 2001) Find the sum of all roots, real and nonreal, of the equation $$x^{2001}+(\frac{1}{2}-x)^{2001}=0$$ given that there are no multiple roots.

Finding Roots

6) A quartic polynomial with rational coefficients has roots $$1 + \sqrt{5}$$ and $$\frac{1}{2}+\frac{i\sqrt{3}}{2}$$. Find all other roots.

7) Find all roots of the equation $$x^3 - 10x^2 + 23x - 14 = 0$$.

Transforming Polynomials

8) Find $$\frac{1}{x}+\frac{1}{y}$$ where $$x$$ and $$y$$ are roots of the polynomial $$x^2+5x+6$$.

9) Find $$\frac{1}{x}+\frac{1}{y}$$ where $$x$$ and $$y$$ are roots of the polynomial $$x^2+11x+19$$

10) Find $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_6}$$ where $$x_1, x_2, ..., x_6$$ are roots of the polynomial $$x^6+12x^5+11x^4+16x^3+24x^2+8x+1$$

Newton Sums

11) Find the sum of the cubes of the solutions of $x^2-3x+3=0$.