Section 1.6Other 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:

4 86 24x x

Example

Solve by Factoring:

4 213 36x x

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.

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.

2

4

If we square both sides, we obtain

x 16

16 -4 or 4

x

x

Example

Solve and check your answers:

5 1x x

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

Graphing Calculator5 1x x

Move all terms to one side. 5 1x x

See the next slide

Press 2nd Window in order to Set up the Table.

Press the Graph key. Look for the zero of the function – the x intercept.

The Graphing Calculator’s Table

Not a solution

Is a solution

5 1x x Press 2nd Graph in order to get the Table.

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:

3 6 6 2x x

Equations with

Rational Exponents

Example

Solve:2

34 8 0x

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:4 213 36 0x x

Example

Simplify:2 1

3 32 10 0x x

Equations Involving

Absolute Value

Example

Solve:

2 4 14x

Absolute Value Graphs

1 4y x

1y x

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.

1 3y x

(a)

(b)

(c)

(d)

2 2 5x x

Solve, and check your solutions:

3

3, 9

3,9

3,9

(a)

(b)

(c)

(d)

Solve:

2 3 17x

5, 3

10, 7

2, 7

2,7