Lesson Entertainment problems on evaluating expressions

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Entertainment problems on evaluating expressions


Problem 1

If   x%5E2 - x + 1 = 0,   find  x%5E2020 + x%5E1010 - 1.

Solution 

Let 

    x%5E2 - x + 1 = 0.    (1)


Multiply both sides by (x+1).  You will get

    x%5E3 + 1 = 0,  or  x%5E3 = -1.


    +---------------------------------------------------------------+
    |    So,  x  is a complex number, which is a cubic root of -1.  |
    +---------------------------------------------------------------+


2020 = 673*3 + 1.  Therefore

     x%5E2020 = %28%28x%5E3%29%5E673%29%2Ax = %28%28-1%29%5E673%29%2Ax= -x.


     
Next, 1010 = 336*3 + 2.  Therefore

     x%5E1010 = %28%28x%5E3%29%5E336%29%2Ax%5E2 = %28%28-1%29%5E336%29%2Ax%5E2 = x%5E2.



Thus,  x%5E2020 + x%5E1010 = -x + x%5E2 = x%5E2 - x.


But, due to (1),   x%5E2 - x = -1.



Therefore,  x%5E2020 + x%5E1010 = -1,  or  x%5E2020 + x%5E1010 + 1 = 0.



Now subtract 2 from both sides of the last equality.  You will get

    x%5E2020 + x%5E1010 - 1 = -2.


At this point, the solution is complete.


ANSWER.  If  x%5E2 - x + 1 = 0,  then  x%5E2020 + x%5E1010 - 1  is equal to  -2.

Problem 2

If   2^a = 3,  3^b = 2,   find   1/(a+1) + 1/(b+1).

Solution 

If 2^a = 3  and  3^b = 2,  then  2^(ab) = (2^a)^b = 3^b = 2,  which implies  ab = 1.


Now,    1%2F%28a%2B1%29 + 1%2F%28b%2B1%29 = %28%28a%2B1%29+%2B+%28b%2B1%29%29%2F%28%28a%2B1%29%2A%28b%2B1%29%29 = 

      = %28a+%2B+1+%2B+b+%2B+1%29%2F%28ab+%2B+a+%2B+b+%2B+1%29 = %28a%2Bb%2B2%29%2F%281+%2B+a+%2B+b+%2B+1%29 = %28a%2Bb%2B2%29%2F%28a%2Bb%2B2%29 = 1.


ANSWER.  If  2^a = 3,  3^b = 2,  then  1%2F%28a%2B1%29 + 1%2F%28b%2B1%29 = 1.

Problem 3

Let  p,  q,  r,  and  s  be the roots of   g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
Compute   1/p + 1/q + 1/r + 1/s.

Solution 

        Notice that since the constant term is not zero, 
           no one root p, q, r  or  s  is zero.


Reduce the given equation to the standard form combining like terms.  You will get

    g(x) = x^4 + 2x^3 + 16x^2 + 20x - 31.    (1)


Divide this reduced equation by x^4.  You will get another polynomial (like a sock turned inside out)

    1 + 2%2Fx + 16%2Fx%5E2 + 20%2Fx%5E3 - 31%2Fx%5E4.    (2)


If some value p, q, r, s  is the root to polynomial (1),
then 1/p, 1/q, 1/r, 1/s  is the zero of function (2).


Let's consider now the polynomial

    h(y) = 1 + 2y + 16y^2 + 20y^3 - 31y^4.    (3)


Compare (3) with (2) and recognize that (3) is the same expression as (2) with replaced  '1/x' by  'y'.


Since p, q, r and s are the roots for polynomial (1),
1/p, 1/q, 1/r and 1/s are the roots for polynomial (3).


Now apply Vieta's theorem and find that the sum of the roots 1/p, 1/q, 1/r, 1/s
is equal to the coefficient 20 at y^3 in polynomial (3), divided by the leading coefficient -31 at y^4,
taken with the opposite sign

    1/p + 1/q + 1/r + 1/s = -20%2F%28-31%29 = 20%2F31.


At this point, the problem is solved in full, without making cumbersome calculations.


ANSWER.  1/p + 1/q + 1/r + 1/s = 20%2F31.

Isn't it beautiful ?


This method is called "turning a polynomial inside out".

Turning a polynomial inside out can be done mentally,
so one can write an answer immediately, without making/writing these reasons on paper.

If you show this focus-pocus to your teacher/professor or at the interview,
the other side will be shocked to see such an elegant solution.


My other lessons on Evaluating expressions in this site are

    - HOW TO evaluate expressions involving  %28x+%2B+1%2Fx%29,  %28x%5E2%2B1%2Fx%5E2%29  and  %28x%5E3%2B1%2Fx%5E3%29
    - Advanced lesson on evaluating expressions
    - HOW TO evaluate functions of roots of a square equation
    - HOW TO evaluate functions of roots of a cubic and quartic equation
    - Problems on Vieta's formulas
    - Advanced problems on Vieta's theorem
    - Miscellaneous problems on Vieta's theorem
    - Evaluating expressions that contain infinitely many square roots
    - Solving equations that contain infinitely many radicals
    - Problems on evaluating in Geometry
    - Evaluating trigonometric expressions
    - Evaluate the sum of the coefficients of a polynomial
    - Miscellaneous evaluating problems
    - Advanced evaluating problems
    - Lowering a degree method
    - Find the number of factorable quadratic polynomials of special form
    - Evaluating a function defined by functional equation
    - Math circle level problems on evaluating expressions
    - Math circle level problems on finding polynomials with prescribed roots
    - Math Olympiad level problem on evaluating a 9-degree polynomial
    - Upper league problem on evaluating the sum
    - Finding coefficients of decomposition of a rational function
    - Upper level problem on evaluating an expression of polynomial roots
    - A truly miraculous evaluating problem with a truly miraculous solution
    - OVERVIEW of lessons on Evaluating expressions

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