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 -Ö2 y) · (y2  + 1 + Ö2 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 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 Any equation where the variable is inside a radical is called an irrational equation.
 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:    Solved problems contents