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# CHAPTER 3 POLYNOMIAL AND RATIONAL FUNCTIONS

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```CHAPTER 3
POLYNOMIAL AND
RATIONAL FUNCTIONS
вЂў Objectives
вЂ“ Recognize characteristics of parabolas
вЂ“ Graph parabolas
вЂ“ Determine a quadratic functionвЂ™s minimum or
maximum value.
вЂ“ Solve problems involving a quadratic
functionвЂ™s minimum or maximum value.
f(x)= ax пЂ« bx пЂ« c
2
graph to be a parabola. The vertex
of the parabolas is at (h,k) and вЂњaвЂќ
describes the вЂњsteepnessвЂќ and
direction of the parabola given
f ( x) пЂЅ a( x пЂ­ h) пЂ« k
2
Minimum (or maximum) function
value for a quadratic occurs at the
vertex.
вЂў If equation is not in standard form, you may have
to complete the square to determine the point
(h,k). If parabola opens up, f(x) has a min., if it
opens down, f(x) has a max.
f ( x) пЂЅ 2 x пЂ­ 4 x пЂ« 3
2
f ( x) пЂЅ 2( x пЂ­ 2 x) пЂ« 3
2
f ( x ) пЂЅ 2 ( x пЂ­ 1) пЂ­ 2 пЂ« 3 пЂЅ 2 ( x пЂ­ 1) пЂ« 1
2
2
( h , k ) пЂЅ (1,1)
вЂў This parabola opens up with a вЂњsteepnessвЂќ of 2
and the minimum is at (1,1). (graph on next page)
Graph of
f ( x ) пЂЅ 2 x пЂ­ 4 x пЂ« 3 пЂЅ 2 ( x пЂ­ 1) пЂ« 1
2
2
3.2 Polynomial Functions & Their
Graphs
вЂў Objectives
вЂ“ Identify polynomial functions.
вЂ“ Recognize characteristics of graphs of
polynomials.
вЂ“ Determine end behavior.
вЂ“ Use factoring to find zeros of polynomials.
вЂ“ Identify zeros & their multiplicities.
вЂ“ Use Intermediate Value Theorem.
вЂ“ Understand relationship between degree &
turning points.
вЂ“ Graph polynomial functions.
f ( x ) пЂЅ a n x пЂ« a n пЂ­1 x
n
n пЂ­1
пЂ« ... пЂ« a 2 x пЂ« a1 x пЂ« a 0
2
вЂў The highest degree in the polynomial is the
degree of the polynomial.
вЂў The leading coefficient is the coefficient of the
highest degreed term.
вЂў Even-degree polynomials have both ends
opening up or opening down.
вЂў Odd-degree polynomials open up on one end and
down on the other end.
вЂў WHY? (plug in large values for x and see!!)
Zeros of polynomials
вЂў When f(x) crosses the x-axis.
вЂў How can you find them?
вЂ“ Let f(x)=0 and solve.
вЂ“ Graph f(x) and see where it crosses the
x-axis.
What if f(x) just touches the x-axis, doesnвЂ™t
cross it, then turns back up (or down) again?
This indicates f(x) did not change from positive
or negative (or vice versa), the zero therefore
exists from a square term (or some even
power). We say this has a multiplicity of 2 (if
squared) or 4 (if raised to the 4th power).
Intermediate Value Theorem
вЂў If f(x) is positive (above the x-axis) at
some point and f(x) is negative (below the
x-axis) at another point, f (x) = 0 (on the
x-axis) at some point between those 2 pts.
вЂў True for any polynomial.
Turning points of a polynomial
вЂў If a polynomial is of degree вЂњnвЂќ, then it has
at most n-1 turning points.
вЂў Graph changes direction at a turning point.
Graph
f ( x ) пЂЅ 2 x пЂ­ 6 x пЂ« 18 x
3
2
f ( x ) пЂЅ 2 x ( x пЂ­ 3 x пЂ« 9 ) пЂЅ 2 x ( x пЂ­ 3)
2
2
Graph, state zeros & end behavior
f ( x ) пЂЅ пЂ­ 2 x пЂ« 12 x пЂ­ 18 x пЂЅ пЂ­ 2 x ( x пЂ­ 6 x пЂ« 9 )
3
f ( x ) пЂЅ пЂ­ 2 x ( x пЂ­ 3)
2
2
2
вЂў END behavior: 3rd degree equation and the leading
coefficient is negative, so if x is a negative number such as
-1000, f(x) would be the negative of a negative number,
which is positive! (f(x) goes UP as you move to the left.)
and if x is a large positive number such as 1000, f(x) would
be the negative of a large positive number (f(x) goes
DOWN as you move to the right.)
вЂў ZEROS: x = 0, x = 3 of multiplicity 2
вЂў Graph on next page
Graph f(x)
f ( x ) пЂЅ пЂ­ 2 x пЂ« 12 x пЂ­ 18 x пЂЅ пЂ­ 2 x ( x пЂ­ 6 x пЂ« 9 )
3
f ( x ) пЂЅ пЂ­ 2 x ( x пЂ­ 3)
2
2
2
Which function could possibly
coincide with this graph?
1) пЂ­ 7 x пЂ« 5 x пЂ« 1
5
2 )9 x пЂ« 5 x пЂ­ 7 x пЂ« 1
5
2
3)3 x пЂ« 2 x пЂ« 1
4
2
4) пЂ­ 4 x пЂ« 2 x пЂ« 1
4
2
3.3 Dividing polynomials;
Remainder and Factor Theorems
вЂў Objectives
вЂ“ Use long division to divide polynomials.
вЂ“ Use synthetic division to divide polynomials.
вЂ“ Evaluate a polynomials using the Remainder
Theorem.
вЂ“ Use the Factor Theorem to solve a polynomial
equation.
How do you divide a polynomial by
another polynomial?
вЂў Perform long division, as you do with
numbers! Remember, division is repeated
subtraction, so each time you have a new
term, you must SUBTRACT it from the
previous term.
вЂў Work from left to right, starting with the
highest degree term.
вЂў Just as with numbers, there may be a
remainder left. The divisor may not go into
the dividend evenly.
Remainders can be useful!
вЂў The remainder theorem states: If the
polynomial f(x) is divided by (x вЂ“ c), then
the remainder is f(c).
вЂў If you can quickly divide, this provides a
nice alternative to evaluating f(c).
Factor Theorem
вЂў f(x) is a polynomial, therefore f(c) = 0 if
and only if x вЂ“ c is a factor of f(x).
вЂў If we know a factor, we know a zero!
вЂў If we know a zero, we know a factor!
```
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