hawkes learning systems: college algebra
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Hawkes Learning Systems: College Algebra. Section 5.1: Introduction to Polynomial Equations and Graphs. Objectives. Zeros of polynomials and solutions of polynomial equations. Graphing factored polynomials. Solving polynomial inequalities. Zeros of a Polynomial. - PowerPoint PPT PresentationTRANSCRIPT
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Hawkes Learning Systems:College Algebra
Section 5.1: Introduction to Polynomial Equations and Graphs
HAWKES LEARNING SYSTEMS
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Copyright © 2011 Hawkes Learning Systems. All rights reserved.
Objectives
o Zeros of polynomials and solutions of polynomial equations.
o Graphing factored polynomials.
o Solving polynomial inequalities.
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Zeros of a Polynomial
The number k is said to be a zero of the polynomial function if . This is also expressed by saying that k is a root or a solution of the equation
Note: k may be a complex number.
11 1 0...n n
n nf x a x a x a x a
0f k
.0f x
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Zeros of a Polynomial
If f is a polynomial with real coefficients and if k is a real number zero of f, then the statement means the graph of f crosses the x-axis at
In this case, may be referred to as an x-intercept of f. .
0f k .,0k
fy
x ,0k
,0k
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Polynomial Equations
A polynomial equation in one variable, say the variable x, is an equation that can be written in the form
where are constants. Assuming , we say such an equation is of degree n.
11 1 0... 0n n
n na x a x a x a
1 1 0n na a a a, ,..., , 0na
For example: 2 3 or 6 3 1 7 .0 0x x x
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2 2 2
Example 1: Zeros of Polynomials and Solutions of Polynomial Equations
Verify that the given value of solves the corresponding polynomial equation.
3 22 12 ; 2x x x x
3 ?22 12
?2 8 4 24
24 24
x
Substitute –2 for x in the original equation.
Simplify, and solve the equation.
Thus, is a solution to the equation. 2x
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Example 2: Zeros of Polynomials and Solutions of Polynomial Equations
Verify that the given value of solves the corresponding polynomial equation.
Although we could verify the solution by substituting for x, it is easier to solve this equation for ourselves using the quadratic formula.
2 3 73 1 4 ; 8ix x x
24 3 1 0x x
23 3 4 4 12 4
x
3 78ix
x
Continued on the next slide…
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Example 2: Zeros of Polynomials and Solutions of Polynomial Equations (Cont.)
One of the two resulting solutions for x is equivalent to the value we were given for x at the beginning of the problem, and thus the given value of x solves the equation.
2 3 73 1 4 ; 8ix x x
3 78ix
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Example 3: Zeros of Polynomials and Solutions of Polynomial Equations
Verify that the given value of x solves the corresponding polynomial equation.
235 ; 0
2xx x
i
?2
3 05
20
i
0 0
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Graphing Factored Polynomials
The behavior of a polynomial function as can be determined as follows:o As , the leading term of
. dominates the behavior.
o If n is even, as , and if n is odd, then . as and as .
o If an is positive, multiplying by an merely compresses or stretches the graph of , while if an is negative, the graph of is the reflection with respect to the x-axis of the graph of .
x
x 1
1 1 0...n nn nf x a x a x a x a
nx x
nx x nx x
nxn
na xn
na x
nx
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Graphing Factored Polynomials
Summary: n even n odd
x x
nx nx nx
nx
No change. is reflected over the x-axis.
na positive na negativen
na x
Note: stretches or compresses the graph. na
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Graphing Factored Polynomials
For the y-intercept is
11 1 0...n n
n nf x a x a x a x a
00, .ay
x
f x
00,a
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Graphing Factored Polynomials
If we are able to factor a given polynomial f into a product of linear factors, every linear factor with real coefficients will correspond to an x-intercept of the graph of f. For example, has the x-intercepts: 3 5 2 2 6 0x x x
52,0 , ,0 ,
3.3,0x
y
x
f x
2,0 5 ,0
3
3,0
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Example 4: Graphing Factored Polynomials
Sketch the graph of the following polynomial function, paying particular attention to the x-intercept(s), the y-intercept, and the behavior as x .
2 1 2f x x x x
1-intercepts: ,0 , 0,0 , 2,0
2x
-intercept: 0,0y
If we were to multiply out the three linear factors of f, the highest degree term would be . The degree of f and the fact that the leading coefficient is negative indicates how f behaves as
32xx .
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Example 5: Graphing Factored Polynomials
Sketch the graph of the following polynomial function, paying particular attention to the x-intercept(s), the y-intercept, and the behavior as x .
4 1g x x
-intercepts: 1,0 , 1,0x
-intercept: 0, 1y
21 1 1g x x x x
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Example 6: Graphing Factored Polynomials
Sketch the graph of the following polynomial function, paying particular attention to the x-intercept(s), the y-intercept, and the behavior as .x
2 2 3h x x x
3 1h x x x
-intercepts: 3,0 , 1,0x
-intercept: 0, 3y
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Solving Polynomial Inequalities
Every polynomial inequality can be rewritten in the form where f is a polynomial function. This will be the key to solving the inequality.
By graphing the polynomial f, we will be able to easily pick out the intervals that solve the inequality.
0 0 0 0f x f x f x f x , , , or ,
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Example 7: Solving Polynomial Inequalities
Solve the following polynomial inequality. 22 3 9x x
22 3 9 0x x 2 3 3 0x x
3-intercepts: ,0 , 3,0
2x
-intercept: 0, 9y 3, 3,2
Now graph the function
using: 2 3 3f x x x
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Example 8: Solving Polynomial Inequalities
Solve the following polynomial inequality. 4 2 32x x x
4 3 22 0x x x
2 2 2 0x x x
2 2 1 0x x x
-intercepts: 2,0 , 0,0 , 1,0x
-intercept: 0,0y 2,0 0,1
Now graph the function
using: 2 2 1f x x x x
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Example 9: Solving Polynomial Inequalities
Solve the following polynomial inequality.
3 1 2 0x x x
-intercepts: 3,0 , 1,0 , 2,0x
-intercept: 0, 6y 3, 1 2,
Graph the function
using: 3 1 2f x x x x