Lectures on Math 10

1. Rational Root Theorem

2. 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)

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

4. Polynomial Inequalities
1. x3
− 3x2
- x- 3 ≥ 0
(x+3)(x+1)(x-1)
Critical Numbers: -3, -1, 1
Conclusion : -3≤x≤-1 OR x≥1

5. 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

6. Graphing
Y= (x+3)(x+1)(x-1)
X intercepts:
-3, -1, 1
Y intercept:
15
Interval Test no. Result
x≤-3 -4 -15
-3≤x≤-1 -2 3
-1≤x≤1 0 -3
x≥1 2 15

8. Rough Sketching

9. Points to follow
For positive odd degree polynomial functions:
The graph will be rising from the left and rising to the
right.

10. Graph : f(x)= (𝑥 + 1) (𝑥 − 2) (𝑥 − 4)
Rising to
the right
Rising from
the left

11. Points to follow
For negative odd degree polynomial function:
The graph will be falling from the left and falling to the
right.

12. Graph : f(x)= -(𝑥 + 3) (𝑥 + 2) (𝑥 − 1)3
Falling from the left
Falling to the right

13. Points to follow
For positive even degree polynomial functions:
The graph will be falling from the left and rising to the
right.

14. Graph : y= 𝑥4
-2𝑥2
-15
Falling from the left Rising to the right

15. Points to follow
For negative even degree polynomial functions:
The graph will be rising from the left and falling to the
right.

16. Graph: f(x)= -x 𝑥 + 5 2
(𝑥 + 3)
Rising
from the
left
Falling to
the right

17. 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.

18. Root or Zero Multiplicity Characteristic
of Multiplicity
Behavior of
graph relative
to x-axis at this
root
-2 2 Even Bounces
-1 3 Odd Crosses
1 4 Even Bounces
2 1 Odd Crosses

19. Example: y= (𝑥 + 2)2(𝑥 + 1)3(𝑥 − 1)4 (𝑥 − 2)