4.2 Polynomial Functions and Models
nth degree polynomial function: f(x)=anxn+an-1xn-1+ ... +a1x+a0.
≠0
f(x)=x3­2x2+7x­6
f(1)=
A number r is called a root (or a zero) of a polynomial function f(x) if f(r)=0.
(roots locate x-intercepts of the graph!)
nth degree polynomial equation: equiv. to anxn+an-1xn-1+ ... +a1x+a0=0.
≠0
"End behavoir" means what happens to the graph as x "goes" toward ±∞.
Even Degree
Odd Degree
Positive Leading Coefficient
Up on both sides Up on the right, down on the left
Negative Leading Coefficient
Down on both sides Down on the right, up on the left Find "end behavior" for:
f(x)= -5x4+7x3+2x2+4x-1
g(x)= 3x3-2x+7
1
Root-Factor Correspondance: r is a root of a polynomial function f(x) if and only if (x-r) is a factor of f(x).
Graph: f(x) = ­(x­1)(x­3)(x­5)
Roots / x-intercepts ?
End behavior ?
A poly. function of degree n may have at most n x-intercepts and (n-1) "turning points."
Intermediate Value Theorem:
If a<b and f(a),f(b) have different signs, for a polynomial f(x), then f(x) has a zero between a and b.
Let f(x)=x4­7x3+4.
Prove f(x) has a real zero in (0,1):
2
The multiplicity of a factor in a polynomial f(x) is the _______ of times it appears as a factor.
the multiplicity of (x-1) is 1
Let f(x)= -(x-1)(x-7)3
the multiplicity of (x-7) is 3
Geometric meaning (this is important): If the multiplicity is even the the graph touches the x­axis but does not cross at the corresponding root. If the multiplicity is odd then the graph crosses the x­axis at the corresponding root.
Graph of f(x)=(x+1)2(x-2)
Incidentally solve the inequality:
(x+1)2(x-2)<0
3
Let f(x)=(x+1)(x-1)2(x-3).
End behavior tells us what?
x and y - intercepts?
Multiplicity tells us what?
Sketch a graph:
Reference points:
4
You try:
1)
Graph f(x)= ­(x+2)(x­3)
2)
Find the end behavior for g(x)=(x­1)11(x­2)22(x­3)33
3)
Let f(x)= -(x+5)2(x-1).
End behavior tells you what?
Multiplicity tells you what?
x and y - intercepts?
Sketch a graph!
5