Lesson OVERVIEW of lessons on Evaluating expressions

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OVERVIEW of lessons on Evaluating expressions


My lessons on Evaluating expressions in this site are

    - HOW TO evaluate expressions involving  %28x+%2B+1%2Fx%29,  %28x%5E2%2B1%2Fx%5E2%29,  %28x%5E3+%2B+1%2Fx%5E3%29  and  %28x%5E5%2B1%2Fx%5E5%29
    - Advanced lesson on evaluating expressions
    - HOW TO evaluate functions of roots of a quadratic 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 problems on evaluating an expression of polynomial roots
    - A truly miraculous evaluating problem with a truly miraculous solution
    - Entertainment problems on evaluating expressions

List of the lessons with short annotations


HOW TO evaluate expressions involving  %28x+%2B+1%2Fx%29,  %28x%5E2+%2B+1%2Fx%5E2%29,  %28x%5E3+%2B+1%2Fx%5E3%29  and  %28x%5E5+%2B+1%2Fx%5E5%29

      Problem 1.  If  x + 1%2Fx = 3,  then find  x%5E2 + 1%2Fx%5E2  and  x%5E3 + 1%2Fx%5E3.

      Problem 2.  If  x + 1%2Fx = 7,  then find  x%5E2 + 1%2Fx%5E2  and  x%5E3 + 1%2Fx%5E3.

      Problem 3.  If  x + 1%2Fx = 7,  then find  sqrt%28x%29 + 1%2Fsqrt%28x%29.

      Problem 4.  If x = 3%2Bsqrt%288%29,  then find  %28x%5E6%2Bx%5E4%2Bx%5E2%2B1%29%2Fx%5E3.

      Problem 5.  If   x + 1%2Fx = 5,   find   x%5E5 + 1%2Fx%5E5.

      Problem 6.  If  x  is the root of the equation  x%5E2+-+3x+%2B+1 = 0,  find   x%5E5 + 1%2Fx%5E5.

      Problem 7.  If   m - 1%2Fm = 5,  find   m%5E2 - 1%2Fm%5E2.


Advanced lesson on evaluating expressions

      Problem 1.  If   a + 1%2F%283a%29 = 2,  then find the value of   9a%5E2 + 1%2Fa%5E2.

      Problem 2.  If   a + 1%2Fa = 2,  find   9a%5E2 + 1%2Fa%5E2.

      Problem 3.  Given the following equalities   g(x) + g(x + 3) = 2x + 5   and   g(2) + g(8) = 12,  find   g(5).

      Problem 4.  If  %28x%2B%281%2Fx%29%29%5E2 = 7  and  x  is a positive real number,  find the exact value of  x%5E3 + 1%2Fx%5E3.

      Problem 5.  If  x%5E2 + 1%2Fx%5E2 = 102,  find the value of  x%5E3 - 1%2Fx%5E3.

      Problem 6.  If   x+y=1   and   x%5E3%2By%5E3=19,  find the value of   x%5E2%2By%5E2.

      Problem 7.  If  x%5E2%2Bx-1 = 0,  what is the value of  x%5E4%2B%281%2Fx%5E4%29 ?


HOW TO evaluate functions of roots of a quadratic equation

      Problem 1.  If  r  and  s  are the roots of the equation   x%5E2+-8x+%2B6 = 0,   then evaluate these expressions:
                           1)  r%5E2+%2B+s%5E2;   2)  r%5E2+%2B+3rs+%2Bs%5E2;   3)  r%5E3+%2B+s%5E3   and   4)  1%2Fr + 1%2Fs.

      Problem 2.  If  r  and  s  are the roots of the equation   x%5E2+%2Bpx+%2Bq = 0,   then evaluate these expressions:
                           1)  r%5E2+%2B+s%5E2;   2)  r%5E2+%2B+3rs+%2Bs%5E2;   3)  r%5E3+%2B+s%5E3   and   4)  1%2Fr + 1%2Fs.

      Problem 3.  If  r  and  s  are the roots of the quadratic equation   3x%5E2+-+4x+%2B+7 = 0,   then find:
                           1)  r%5E2+%2B+s%5E2,  2)  r%5E2+%2B+3rs+%2B+s%5E2,   3)  r%2Fs + s%2Fr   and   4)  1%2Fr%5E2 + 1%2Fs%5E2.


HOW TO evaluate functions of roots of a cubic and quartic equation

      Problem 1.  If  r,  s  and  t  are the roots of the cubic equation   x%5E3-7x%2B5 = 0,   find:
                           1)  r + s + t     2)  rs + rt + st;     3)  rst;     4)  r%5E2+%2B+s%5E2+%2B+t%5E2;     5)  1%2Fr+%2B+1%2Fs+%2B+1%2Ft.

      Problem 2.  If  r,  s  and  t  are the roots of the cubic equation   2x%5E3-7x%2B5 = 0,   find:
                           1)  r + s + t     2)  rs + rt + st;     3)  rst;     4)  r%5E2+%2B+s%5E2+%2B+t%5E2;     5)  1%2Fr+%2B+1%2Fs+%2B+1%2Ft.

      Problem 3.  If  r,  s,  t  and  u  are the roots of the quartic equation   x%5E4-3x%5E3+%2B+5x%5E2+-+7x+%2B+9 = 0,   find:
                           1)  r + s + t + u     2)  rs + rt + ru + st + su + tu;     3)  rst + stu + rtu + rsu;     4)  rstu;     5)  r%5E2+%2B+s%5E2+%2B+t%5E2+%2B+u%5E2;   6)  1%2Fr+%2B+1%2Fs+%2B+1%2Ft+%2B+1%2Fu.


Problems on Vieta's formulas

      Problem 1.  If  x  is the root of the equation   x%5E2+-+3x+%2B+1 = 0,  find   x%5E5 + 1%2Fx%5E5.

      Problem 2.  If the sum of the reciprocals of the roots of the quadratic equation   3x^2 + 7x + k = 0   is  -7/3,  what is  k?

      Problem 3.  If  p  and  q  are the roots of the equation  2x%5E2-x-4=0,  find the equation whose roots are  p-%28q%2Fp%29  and  q-%28p%2Fq%29.

      Problem 4.  Consider polynomial   P(z) = z^5 - 10z^2 + 15z - 6   of complex variable  z.
                         Find the sum and the product of the roots of  P(z).

      Problem 5.  Given that   x%5E2-3x%2B2   is a factor of   x%5E4+%2B+kx%5E3+-+10x%5E2+-+20x%2B24,   evaluate
                         the sum of the four roots of the equation   x%5E4+%2B+kx%5E3+-+10x%5E2+-+20x%2B24+=+0.


Advanced problems on Vieta's theorem

      Problem 1.  One of the roots of the equation  2x%5E2+%2B+1 = h-5x  is four times to other root.  Find the value oh  h.

      Problem 2.  The horizontal line  y = k  intersects the parabola with equation  y = 2(x-3)(x-5)  at points  A  and  B.
                         If the length of line segment  AB  is  6,  what is the value of  k?

      Problem 3.  Find the sum of the squares of the roots to the equation  (4x^2 - 9)^4 - 10(4x^2 - 9)^2 + 9 = 0.

      Problem 4.  If the solution for  x  of   x%5E2%2Bpx%2Bq = 0   are the cubes of the solutions for   x%5E2%2Bmx%2Bn = 0,   express  p  and  q  in terms of  m  and  n.

      Problem 5.  Circle T intersects the hyperbola  y = 1%2Fx  at  (1,1),  (3, 1%2F3),  and two other points.
                         What is the product of the y-coordinates of the other two points?

      Problem 6.  Let  x,  y,  and  z  be real numbers such that  x + y + z,  xy + xz + yz,  and  xyz  are all positive.
                         Prove that  x,  y,  and  z  are all positive.

      Problem 7.  Determine  (r + s)(s + t)(t + r),  if  r,  s,  and  t  are the three real roots of the polynomial   x%5E3+%2B+9x%5E2+-+9x+-+8.

      Problem 8.  Solve equation   2x^3 + 3x^2 + hx + k = 0   and find the values of  h  and  k,  given that  -3  is the first root and the third root is twice the second.

      Problem 9.  Find the quadratic equation such that each of its roots is the sum of a root
                         and its reciprocal of the quadratic equation   2x^2 + 3x + 4 = 0.

      Problem 10.  Write an equation each of whose roots are  2  less than  3  times the roots of   3x^3 + 10x^2 + 7x - 10 = 0.


Miscellaneous problems on Vieta's theorem

      Problem 1.  Let  'a'  and  'b'  be the roots of the quadratic equation   5x^2 - 23x - 4 = 0.
                         Compute   1/a^2 + 1/b^2.

      Problem 2.  Let  'a'  and ' b'  be the solutions to   6x^2 - 16x + 7 = 0.
                         Find   a^2/b + b^2/a.

      Problem 3.  Let  'a'  and  'b'  be the roots of the quadratic   x^2 - 5x + 3 = 0.
                         Find the quadratic whose roots are   a^2/b and b^2/a.

      Problem 4.  There are integers  'b',  'c'  for which both roots of the polynomial   x^2 - x - 3
                         are also roots of the polynomial   x^3 - bx^2 - c.   Determine the ordered pair  (b,c).

      Problem 5.  Let  'p',  'q',  'r',  and  's'  be the roots of   g(x) = x^4 + 2x^3 + 16x^2 + 20x - 31.
                         Compute   p^2*qrs + pq^2*rs + pqr^2*s + pqrs^2.

      Problem 6.  Let  'p',  'q',  'r',  and  's' be the roots of   x^4 + 2x^3 + 16x^2 + 20x - 31.
                         Compute   p^2 + q^2 + r^2 + s^2.


Evaluating expressions that contain infinitely many square roots

      Problem 1.  Simplfy/evaluate   sqrt%285+%2B+sqrt%285+%2B+sqrt%285+%2B+ellipsis%29%29%29.

      Problem 2.  Simplfy   sqrt%282+%2B+sqrt%282+%2B+sqrt%282+%2B+ellipsis%29%29%29.

      Problem 3.  Simplfy   sqrt%286+%2B+sqrt%286+%2B+sqrt%286+%2B+ellipsis%29%29%29.

      Problem 4.  Simplfy   sqrt%2812+%2B+sqrt%2812+%2B+sqrt%2812+%2B+ellipsis%29%29%29.

      Problem 5.  Evaluate   sqrt%287%2F3+%2B+sqrt%287%2F9+%2B+sqrt%287%2F3+%2B+sqrt%287%2F9%29+%2B+ellipsis%29%29%29.

      Problem 6.  Evaluate the value   sqrt%285%2Bsqrt%285%5E2%2Bsqrt%285%5E4%2Bsqrt%285%5E8%2Bellipsis%29%29%29%29.


Solving equations that contain infinitely many radicals

      Problem 1.  Solve an equation sqrt%28y%2Bsqrt%28y%2Bsqrt%28y%2Bellipsis%29%29%29 = 5.

      Problem 2.  Let   a = root%283%2C+a%5E2+%2B+root%283%2C+a%5E2+%2B+root%283%2C+a%5E2+%2B+ellipsis%29%29%29.   Prove that   a = root%287%2C+4a%5E4+%2B+root%287%2C+4a%5E4+%2B+root%287%2C+4a%5E4+%2B+ellipsis%29%29%29.


Problems on evaluating in Geometry

      Problem 1.  A cuboid has latent faces and top/bottom faces of areas  10 cm^2,  7.5 cm^2  and  3 cm^2,  respectively.
                         What is the volume of the solid in cm^3 ?

      Problem 2.  Consider right rectangular prism.  The total surface area of the prism is  1 square unit.
                         Also,  the sum of all the edges of the prism is  8  units.
                         Find the length of the diagonal joining one corner of the prism to the opposite corner.


Evaluating trigonometric expressions

      Problem 1.  What is the value of  cos(2a),  if  sin(3a) = 2*sin(a)?

      Problem 2.  What is the value of  cos(2a) ,  if  2*cos(3a)= cos(a)?

      Problem 3.  If sin%5E2%28pi%2F9%29 + sin%5E2%282pi%2F9%29 + sin%5E2%283pi%2F9%29 + sin%5E2%284pi%2F9%29 = 9/4,

                         evaluate cos%5E2%28pi%2F9%29 + cos%5E2%282pi%2F9%29 + cos%5E2%283pi%2F9%29 + cos%5E2%284pi%2F9%29.

      Problem 4.  Find value of   cos(2π/7) + cos(4π/7) + cos(6π/7).

      Problem 5.  If   sin⁶β + cos⁶β = ¼,   find (1/sin⁶β) + (1/cos⁶β).

      Problem 6.  If   8*cos(a) - 8*sin(a) = 3,   find   55%2Atan%28a%29+ + 55%2Ftan%28a%29.


Evaluate the sum of the coefficients of a polynomial

      Problem 1.  If   F(x) = ax^2+bx+c,  and   F(x+5) = x^2+9x-7,  what is the sum of   a+b+c?

      Problem 2.  A quartic polynomial in the form   f(x) = ax^4 + bx^3 + cx^2 + dx + e   is such that the coefficient
                         of the quadratic and linear terms are  10  and  -18  respectively.  Additionally,  f(0)=  9  and  x= 1  is a root.   What is the value of  (a + b)?

      Problem 3.  Let  f(x)  be a quadratic polynomial such that  f(-4) = -22,  f(-1)=2,  and  f(2)=-1.  Let  g(x) = f(x)^{16}.
                         Find the sum of the coefficients of the terms in  g(x)  with even exponents.


Miscellaneous evaluating problems

      Problem 1.  If  a-b = 4  and  ab+c^2+4 = 0,  find a+b.

      Problem 2.  If   (2006-a)(2004-a)=2005,  find (2006-a)^2 + (2004-a)^2.

      Problem 3.  If  a-b = -6  and  b+c = 9,   find the value of   3a%5E2-b%5E2-2c%5E2.

      Problem 4.  Evaluate the following sums

                         a)   1 - 3 + 5 - 7 + 9 -11 + ... + 201 - 203

                         b)   300 - 299 + 298 - 297 + ... + 100 - 99

      Problem 5.  Express the following product as reduced fraction   %281-1%2F2%29%281-1%2F3%29%281-1%2F4%29%2Aellipsis%2A%281-1%2F2008%29.

      Problem 6.  Find the value of the product   %281-1%2F2%5E2%29%281-1%2F3%5E2%29%281-1%2F4%5E2%29%2Aellipsis%2A%281-1%2F999%5E2%29.

      Problem 7.  Find the sum of the sequences  8,  88,  888,  . . .  up to  n  terms.

      Problem 8.  Find the sum of all possible  4-digit numbers that can be formed using the digits  2,  4,  5,  6,  7,  and  8,  with no repeated digits.

      Problem 9.  If   x + y + z = 20  and   x^2 + y^2 + z^2 = 100,  find the value of   xy + xz + yz.

      Problem 10.  If   F(x) = ax^2 +bx+c   and   F(x+5) = x^2+9x-7,   find the sum of   a+b+c.

      Problem 11.  For positive numbers  a,  b,  and c,  if 2ab = 1,  3bc = 2,  and  4ca = 3,  find the value of  a + b + c.

      Problem 12.  All students in  Ms.  Fay's  Spanish class are either going on the  Spain trip,  the  Mexico trip,  or both.
                         1/4 of the students going to  Spain are also going to  Mexico,
                         and  2/7  of the students going on the  Mexico trip are also going to  Spain.
                         Which of the following could be the total number of students in  Ms.  Fay's  Spanish class?
                             (A)   26;     (B)   27;     (C)   28;     (D)   29;     (E)   30.

      Problem 13.  If   2%5Ex = 4%5Ey = 8%5Ez   and   1%2F2x + 1%2F4y + 1%2F8z = 22%2F7,   show that   x = 7%2F16,  y = 7%2F32  and  z = 7%2F48.


Advanced evaluating problems

      Problem 1.  Let  x  be a real number such that   625%5Ex = 64.
                         Then   125%5Ex = a%2Asqrt%28b%29.   Find  " a "  and  " b ".

      Problem 2.  If  x%5E2+%2B+y%5E2+%2B+z%5E2 = 2%2A%285X+-+8Y+%2B+6Z%29+-+125,  where  x,  y  and  z  are real numbers,  find the value of  x+y+z.

      Problem 3.  If   f(x) = 9%5Ex%2F%283+%2B+9%5Ex%29,   prove that   f%281%2F2016%29 + f%282%2F2016%29 + f%283%2F2016%29 + . . . + f%282015%2F2016%29 = 2015%2F2.

      Problem 4.  If  1%2Fx - 1%2Fy = 1%2F%28x%2By%29,   find   y%2Fx - x%2Fy.

      Problem 5.  If   1%2Fx - 1%2Fy = 1%2F%28x%2By%29,   find   y%2Fx + x%2Fy.

      Problem 6.  The lengths of the sides of an equilateral triangle are  log%284%2Ca%29,  log%2810%2Cb%29,  log%2825%2C%28a%2Bb%29%29,
                         where  "a"  and  "b"  are positive numbers.  What is the value of  a%2Fb?

      Problem 7.  If   (1 + a)(1 + b)(a + b) = 1530   and   a^3 + b^3 = 1241,   find (a + b).


Lowering a degree method

      Problem 1.  If  "a"  and  "b"  are the roots of equation   x%5E2 = x+1,  find the value of   a%5E5 + b%5E5.

      Problem 2.  If   x^3 + 5x - 10 = 0,   then find the value of   x^7 + 100x^2 + 25x.

      Problem 3.  Suppose x is a positive number such that   x%5E2+=+1-x.
                         There is a unique choice of whole numbers  p  and  q  so that   x%5E8=p-qx.  Find  p+q.


Find the number of factorable quadratic polynomials of special form

      Problem 1.  Find the number of positive integers  n,   1 <= n <= 1000,   for which
                         the polynomial   x%5E2+%2B+x+-+n   can be factored as the product of two linear binomials with integer coefficients.


Evaluating a function defined by functional equation

      Problem 1.  The function  f  satisfies   f%28sqrt%282x+-+1%29%29 = 1%2F%282x+-+1%29
                         for all  x  not equal to  1/2.   Find f(2).

      Problem 2.  Suppose a function  f  is such that   f(1/x) - 3f(x) = x   for every non-zero  x.   Find  f(2).

      Problem 3.  If   f%283x%2F%28x-4%29%29 = x%5E2+%2B+x+%2B+1,   what is the value of  f(5)?

      Problem 4.  If   f(x) = 1/(1 - x) ,   find   (f(f(f(f...f)(sqrt2),  (45 times).


Math circle level problems on evaluating expressions

      Problem 1.  If   f(2a-b) = f(a)*f(b)  for all  "a"  and  "b",  and the function is never equal to zero,
                         find the value of  f(5).

      Problem 2.  A function  f  is defined for integers  m  and  n  as given:
                             f(mn) = f(m)f(n)-f(m+n)+ 1001, where either m or n is equal to 1,  and  f(1)=2.
                         a) Prove that  f(x) = f(x-1) + 1001.
                         b) Find the value of  f(9999).

      Problem 3.  Given that   f%28x%29=5x%5E2-3x%2B7   and   f%28g%28x%29%29=%285x%5E4%2F9%29%2B%2817x%5E2%2F3%29%2B21,   find all possible values
                         for the sum of the coefficients in the quadratic function  g(x).

      Problem 4.  Five different positive integers added two at a time give the following sums:  16,  20,  22,  23,  25,  28,  29,  30,  34  and  37.
                         Find the product of the five integers.

      Problem 5.  There are  5  sacks,  and they are weighed  2  at a time.
                         Their weights are  11,  11.2,  11.3,  11.4,  11.5,  11.6,  11.7,  11.8,  12 and  12.1.
                         This is the weight of all the possible outcomes.  How heavy are each of the sacks?


Math circle level problems on finding polynomials with prescribed roots

      Problem 1.  Find the polynomial with roots  alpha,  beta  and  gamma,  if  alpha%2Abeta%2Agamma = 6,  alpha+%2B+beta+%2B+gamma = 5,  and  alpha%5E2+%2B+beta%5E2+%2B+gamma%5E2 =21.

      Problem 2.  Let the roots of the equation   x^3 -2x^2 -3x-7=0   be  r,  s,  and  t.
                         Find an equation whose roots are  r^2,  s^2  and  t^2.

      Problem 3.  The roots of the polynomial equation  2x%5E3+-+8x%5E2+%2B+3x+%2B+5 = 0  are  alpha,  beta  and  gamma.
                         Find the polynomial equation with roots  alpha%5E2,  beta%5E2,  gamma%5E2.

      Problem 4.  Use this identify

                         tan4Q = %284tanQ-4%28tanQ%29%5E3%29%2F%281-6%28tanQ%29%5E2%2B%28tanQ%29%5E4%29

                         to find the polynomial of least degree that has zeroes  %28tan%28pi%2F24%29%29%5E2,  %28tan%287pi%2F24%29%29%5E2,  %28tan%2813pi%2F24%29%29%5E2,  %28tan%2819pi%2F24%29%29%5E2%29.


Math Olympiad level problem on evaluating a 9-degree polynomial

      Problem 1.  Let   P(x) = 2009x%5E9 + a%5B1%5Dx%5E8 + . . . + a%5B9%5D   such that   P%281%2Fn%29 = 1%2F%28n%5E3%29,  n = 1, 2, . . . , 9.
                         Find   P%281%2F10%29.


Upper league problem on evaluating the sum

      Problem 1.  If   f(n) = log%28%28n%29%29%2Flog%28%282006n-n%5E2%29%29,    find   f(1) + f(2) + f(3) + . . . + f(2005).

Finding coefficients of decomposition of a rational function

      Problem 1.  Find  A,  B  and  C  if   A%2F%28x-1%29 + B%2F%28x-2%29 + C%2F%28x-3%29 = %282x%5E2-6x%2B6%29%2F%28x-1%29%28x-2%29%28x-3%29


Upper level problems on evaluating an expression of polynomial roots

      Problem 1.  If  a,  b,  c   (where  a,  b,  c =/= 0)   are the roots of the equation   x%5E3+%2B+px%5E2+%2B+qx+%2B+r = 0,
                         where  p,  q  and  𝑟 (=/= 0)  are real numbers,  express   1%2Fa%5E3 + 1%2Fb%5E3 + 1%2Fc%5E3   in terms of  p,  q  and  r.

      Problem 2.  If   x%5E5 = 1   with x =/= 1,  find the value of 1%2F%281%2Bx%5E2%29 + 1%2F%281%2Bx%5E4%29 + 1%2F%281%2Bx%29 + 1%2F%281%2Bx%5E3%29 .


A truly miraculous evaluating problem with a truly miraculous solution

      Problem 1.  Given   (x² + 1)(y² + 1) + 25 = 10(x + y),   find (x³ + y³)/220.


Entertainment problems on evaluating expressions

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

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


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