hawkes learning systems: college algebra

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Hawkes Learning Systems:College Algebra

Section 5.1: Introduction to Polynomial Equations and Graphs




Objectives

o Zeros of polynomials and solutions of polynomial equations.

o Graphing factored polynomials.

o Solving polynomial inequalities.




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




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




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




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





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…




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





Verify that the given value of x solves the corresponding polynomial equation.

235 ; 0

2xx x

i

?2

3 05

20

i

0 0




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





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





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





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




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 .





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





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




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 ,




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





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





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

hawkes learning systems: college algebra

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