Transcript x - 7

ALGEBRA
1.4 Polynomials
1.5 Equations
1.4
A polynomial is an algebraic expression that has many terms
connected by only the operations of +, -, and • of variables.
2x + 5
2x3y4z
5 x  7x 19
5x2
x 2
x
2
- 7x + 19
1
2
x
2x
The terms are the parts of the expression that are added or
subtracted.
Number of Terms
Name
Example
1
monomial
3x2y5
2
binomial
3x2 + y5
3
trinomial
3 - x2 + y5
4 or more
no special name
3x5 - 2x4 + 5x3 - 6x2 + 2x
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1.4
The degree of a polynomial is the maximum number of variables that
appear as factors in any one term.
Degree
Name
Example
Memory Aid
0
constant
4
1st
linear
4x;
2nd
quadratic
4x2; 4x2+5x+2 A square is a quadrilateral
3rd
4th
cubic
quartic
4x3; 4xyz+2
5th
quintic
4x5; 3x+4tx2y2 Quintuplets are 5 children
6th or more no special name
Constants do not vary
4x+4
A line has one dimension
A cube has 3 dimensions
4x4; 8x2y2+4x3 A quarter is a fourth of a dollar
4x44
2
1.4
3x2
3 is the numerical coefficient
x is the base
2 is the exponent
x2 is the power
Zero could have many different degrees, therefore it is considered to
be a polynomial with no degree.
0 = 0x = 0x2 = 0xy = 0x92 = 0x3y9z999
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1.4
Multiplying binomials can be done by two different procedures.
1. FOIL
2. Double use of the distributive property
(2x - 3)(3x + 1)
FOIL mean First Outside Inside Last
(2x - 3)(3x + 1)
6x2 + 2x - 9x - 3
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1.4
Multiplying binomials can be done by two different procedures.
1. FOIL
2. Double use of the distributive property
(2x - 3)(3x + 1)
To proceed through double use of the distributive property, think of (2x - 3)
as one number,and distribute this number over the other binomial.
(2x - 3) • 3x + (2x - 3) • 1
Then use the commutative property of multiplication to get
3x(2x - 3) + 1(2x - 3)
Now distribute each of these new values
6x2 - 9x + 2x - 3
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1.4
In both cases, the polynomial can be simplified by combining like terms.
6x2 - 7x - 3
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1.5
Equations are two expressions connected by an equal sign.
Solving an equation means writing its solution set.
Solve: x2 = 25
There are two answers, x = 5 and x = -5, so the solution is
S = {-5, 5}
Solve: 5x - 8 = 2x + 4
3x = 12
x=4
This equation is equivalent to the original one.
S = {4} They have the same solution set.
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1.5
Before writing a solution set, you should check your answer by
substituting the value(s) back into the original equation.
An equation is not solved until you write the solution set.
Adding (or subtracting) both members of an equation by the same
number is justified by the Addition Property of Equality.
Multiplying (or dividing) both members of an equation by the same
number is justified by the Multiplication Property of Equality.
While it is not necessary to justify each step when solving an equation,
it will be necessary to do so when completing algebraic proofs therefore learn the properties.
No Work = No Credit, and the more work you show, the more partial
credit you can earn.
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1.5
Given the equation (x - 3)(x - 7) = 0:
S = {3, 7}
1. Write the solution set.
This equation contains a product which equals zero.
The only way a product of two real numbers can equal zero is for
one of the factors to equal zero.
This fact is expressed as the converse of the Multiplication Property
of Zero.
x - 3 = 0 or x - 7 = 0
x = 3 or x = 7
S = {3, 7}
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1.5
Given the equation (x - 3)(x - 7) = 0:
x-7=0
S = {3, 7}
S = {7}
2. Divide both members of the equation by (x - 3), and write the
solution set for the transformed equation.
(x - 3)(x - 7) = 0 .
(x - 3)
(x - 3)
x-7=0
S = {7}
3. Is the transformed equation equivalent to the original equation?
Explain.
No, the solution sets are different.
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1.5
Given the equation (x - 3)(x - 7) = 0:
x-7=0
(x - 3)(x - 7)(x - 2) = 0
S = {3, 7}
S = {7}
S = {2, 3, 7}
4. Multiply both members of the original equation by (x - 2), then solve
it.
(x - 3)(x - 7)(x - 2) = 0(x - 2)
(x - 3)(x - 7)(x - 2) = 0
x - 3 = 0 or x - 7 = 0 or x - 2 = 0
x = 3 or x = 7 or x = 2
S = {2, 3, 7}
5. What number is a solution for the transformed equation that was
not a solution of the original one?
The new solution is 2.
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1.5
Given the equation (x - 3)(x - 7) = 0:
x-7=0
(x - 3)(x - 7)(x - 2) = 0
S = {3, 7}
S = {7}
S = {2, 3, 7}
6. How can all three of these equations be equal to zero?
Since x is a variable, there is the possibility that any of the factors
could equal zero.
When you multiply or divide by zero, you are performing an
operation which seems perfectly correct, but which changes the
solution set.
Since it is the original equation you are trying to solve, you must be
aware of these issues.
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1.5
You should never divide an equation by something that can equal zero.
DIVISION BY ZERO IS UNDEFINED and therefore NOT ALLOWED!!!!!
When you divide by zero you can lose a valid solutions.
When you multiply by zero, you are performing an irreversible step.
This can lead to an extraneous solution.
An extraneous solution is a solution which satisfies the transformed
equation, but not the original equation. It is an extra solution.
You will find which solutions are extraneous by checking your answer.
You can have equations with no solutions.
In this case the solution set is empty.
S = Ø or S = { }
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1.5
Solve: | x - 2 | = 3
Remember: absolute value is distance from zero.
| x - 2 | = 3 means (x - 2) is three units form zero. This is three units to
the left and three units to the right.
x - 2 = 3 or x - 2 = -3
x = 5 or x = -1
S = {-1, 5}
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