section 1.6 other types of equations. polynomial equations
TRANSCRIPT
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Section 1.6Other Types of Equations
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Polynomial Equations
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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.
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Example
Solve by Factoring:
4 86 24x x
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Example
Solve by Factoring:
4 213 36x x
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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
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Radical Equations
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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.
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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
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Example
Solve and check your answers:
5 1x x
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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.
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The Graphing Calculator’s Table
Not a solution
Is a solution
5 1x x Press 2nd Graph in order to get the Table.
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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.
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Example
Solve:
3 6 6 2x x
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Equations with
Rational Exponents
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Example
Solve:2
34 8 0x
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Equations That Are
Quadratic in Form
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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.
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Example
Simplify:4 213 36 0x x
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Example
Simplify:2 1
3 32 10 0x x
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Equations Involving
Absolute Value
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Example
Solve:
2 4 14x
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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
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(a)
(b)
(c)
(d)
2 2 5x x
Solve, and check your solutions:
3
3, 9
3,9
3,9
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(a)
(b)
(c)
(d)
Solve:
2 3 17x
5, 3
10, 7
2, 7
2,7