Mathematics 10 (Quarter Two)
- 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
- 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)