Transcript Section 1.6 Other Types of Equations Polynomial Equations A polynomial equation is the result of setting two polynomials equal to each other.

Section 1.6
Other Types of Equations
Polynomial Equations
A polynomial equation is the result of setting two
polynomials equal to each other. The equation is in
general form if one side is 0 and the polynomial on
the other side is in descending powers of the variable.
The degree of a polynomial equation is the same as the
highest degree of any term in the equation. Here are
examples of some polynomial equations.
Example
Solve by Factoring:
6 x 4  24 x8
Example
Solve by Factoring:
x 4  13x 2  36
Graphing Equations
You can find the solutions on the graphing calculator for
the previous problem by moving all terms to one side, and
graphing the equation. The zeros of the function are the
solutions to the problem. X4-13X2+36=0
Radical Equations
A radical equation is an equation in which the variable
occurs in a square root, cube root, or any higher root.
We solve the equation by squaring both sides.
x4
If we square both sides, we obtain
x 2  16
x   16  -4 or 4
This new equation has two solutions, -4 and 4. By
contrast, only 4 is a solution of the original
equation, x=4. For this reason, when raising both
sides of an equation to an even power, check
proposed solutions in the original equation.
Extra solutions may be introduced when you raise
both sides of a radical equation to an even power.
Such solutions, which are not solutions of the
given equation are called extraneous solutions or
extraneous roots.
Example
Solve and check your answers:
x  5  x 1
Graphing Calculator
Move all terms to one side.
x  5  x 1
Press Y= to type in the
equation. For the negative use
the white key in the bottom right
hand side. For the use X use
X,T,,,,n

Press the Graph key.
Look for the zero of
the function – the x
intercept.
Press 2nd Window in
order to Set up the Table.
See the next slide
x  5  x 1
The Graphing Calculator’s Table
Press 2nd Graph in order
to get the Table.
Not a solution
Is a solution
x  5  x 1
Solving an Equation That Has Two Radicals
1. Isolate a radical on one side.
2. Square both sides.
3. Repeat Step 1: Isolate the remaining radical on
one side.
4. Repeat Step2: Square both sides.
5. Solve the resulting equation
6. Check the proposed solutions in the original
equations.
Example
Solve:
3x  6  x  6  2
Equations with
Rational Exponents
Example
Solve:
2
3
4x  8  0
Equations That Are
Quadratic in Form
Some equations that are not quadratic can be written as
quadratic equations using an appropriate substitution. Here
are some examples:
An equation that is quadratic in form is one that can be
expressed as a quadratic equation using an appropriate
substitution.
Example
Simplify:
x  13x  36  0
4
2
Example
Simplify:
2
3
1
3
2 x  x  10  0
Equations Involving
Absolute Value
Example
Solve:
2x  4  14
Absolute Value Graphs
y  x 1
y  x 1  3
y  x 1  4
The graph may intersect the x axis at one point, no points or two
points. Thus the equations could have one, or two solutions or no
solutions.
Solve, and check your solutions:
2x  2  x  5
(a)
3
3, 9
(c) 3,9
(b)
(d)
3,9
Solve:
2x  3  17
5, 3
(b) 10, 7
(c) 2, 7
(a)
(d)
2, 7