ALGEBRA - solved problems
  Polynomial and/or polynomial functions and equations
Binomial equations
An equation of the form  axn + b = 0,   a > 0,  b > 0 and  n is a natural number is called the binomial
equation.
Solving binomial equations
By using substitution   the equation transforms to   or y n  1 = 0.
Recall that the expression on the left side can be factorized, for example:
              y3  + 1 = (y + 1) (y2 - y + 1)
              y3  - 1 = (y - 1) (y2 + y + 1)
              y4  + 1 = (y2 + 1)2 - 2y2 = (y2  + 1 -2 y) (y2  + 1 + 2 y)
              y4  - 1 = (y2 - 1) (y2 + 1) = (y - 1) (y + 1) (y2 + 1)
              y5  + 1 = (y + 1)   (y4 - y3 + y2 - y + 1)
              y5  - 1 = (y - 1)   (y4 + y3 + y2 + y + 1)
              y6  - 1 = ( y3  - 1)   (y3  + 1) = (y - 1) (y + 1) (y2 + y + 1) (y2 - y + 1) and so on. 
146.
    Solve the binomial equation  8x3 - 27  = 0.
Solution:   Let substitute     
To obtain the values of the original variable plug the solutions into the substitution  x = (3/2)y
147.
    Solve the binomial equation  x4 + 81 = 0.
Solution:   By substituting  obtained is  
To obtain the values of the original variable plug the solutions into the substitution  x = 3y
Equations reducible to quadratic form - biquadratic equations
Solving biquadratic equations or equations reducible to quadratic
Thus, a biquadratic equation  ax4 + bx2 + c = 0   we can write   a(x2)2 + bx2 + c = 0
and solve as the quadratic equation in the unknown x2 using the substitution  x2 = y.
148.
    Solve the biquadratic equation  3x4 - 4x2 + 1 = 0.
Solution:   By substituting  x2 = y  we get the quadratic equation  3y2 - 4y + 1 = 0
To obtain the values of the original variable plug the solutions into the substitution  x2 = y
149.
    Solve the equation  x6 - 7x3 - 8 = 0  that is reducible to quadratic form.
Solution:   By substituting  x3 = y  we get the quadratic equation  y2 - 7y  - 8 = 0
To obtain the values of the original variable plug the solutions into the substitution  x3 = y
Radical or irrational equations
Any equation where the variable is inside a radical is called an irrational equation.
Solving irrational or radical equations
150.
   Solve the irrational equation
Solution:   First isolate the radical and then square both sides of the equation,
Checking for the solutions:
151.
   Solve the irrational equation
Solution:
Checking for the solutions:
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