Download - Mathematics 10 (Quarter Two)
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=