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