Rational Root Theorem

What is the rational root theorem?

The polynomial equation is p(x)=0 has integral coefficients. If P/Q is a rational root of the polynomial equation. Then P is a factor of Ao and Q is a factor of An (leading coefficient)

P: +1, -1, +3, -3 Q: +1, -1, +2, -2 P/Q: +1, -1, +3, -3, +1/2, -1/2, +3 2, -3/2

Polynomial Inequalities

1. - x- 3 ≥ 0

(x+3)(x+1)(x-1)

Critical Numbers: -3, -1, 1

Conclusion : -3≤x≤-1 OR x≥1

How?

Interval Test No. Result Yes or No

x≤-3 -4 -15 No

-3≤x≤-1 -2 3 Yes

-1≤x≤1 0 -3 No

x≥1 2 15 Yes

Graphing

Y= (x+3)(x+1)(x-1)

X intercepts:

-3, -1, 1

Y intercept:

15

Interval Test no. Resultx≤-3 -4 -15

-3≤x≤-1 -2 3

-1≤x≤1 0 -3

x≥1 2 15

Rough Sketching

Points to follow

For positive odd degree polynomial functions:The graph will be rising from the left and rising to the

right.

Graph : f(x)=

Rising to the right

Rising from the left

Points to follow

For negative odd degree polynomial function:The graph will be falling from the left and falling to the

right.

Graph : f(x)= -

Falling from the left

Falling to the right

Points to follow

For positive even degree polynomial functions:The graph will be falling from the left and rising to the

right.

Graph : y= --15

Falling from the left Rising to the right

Points to follow

For negative even degree polynomial functions:The graph will be rising from the left and falling to the

right.

Graph: f(x)= -x

Rising from the left Falling to

the right

Points to follow

The characteristic of multiplicity ( odd or even ) will determine the behavior of the graph relative to x-axis at the given root.

If the characteristic of multiplicity is odd, it will cross the x-axis.

If the characteristic of multiplicity is even, it will bounce at the x-axis.

Root or Zero Multiplicity Characteristic of Multiplicity

Behavior of graph relative

to x-axis at this root

-2 2 Even Bounces-1 3 Odd Crosses1 4 Even Bounces2 1 Odd Crosses

Example: y=