Lesson 2-6

1. Lesson 2-6 Solving Polynomial Equations by Factoring – Part 2

2. Objective:

3. To solve polynomial equations by various methods of factoring, including the use of the rational root theorem. Objective:

4. When trying to factor a quadratic into two binomials, we only ever concern ourselves with the factors of the a (leading coefficient) and c (constant term).

5. Solve:

6. Solve: 3x2 – 11x – 4 = 0

7. Solve: 3x2 – 11x – 4 = 0 (3x + 1)(x – 4) = 0 Solving for x  x = - 1/3 or x = 4

8. Solve: 3x2 – 11x – 4 = 0 (3x + 1)(x – 4) = 0 Solving for x  x = - 1/3 or x = 4 So we only concerned ourselves with the factors of 3 and 4.

9. We call the possible factors of c  p values.

10. We call the possible factors of c  p values. We call the possible factors of a  q values.

11. This leads us into what is called the Rational Roots Theorem.

12. This leads us into what is called the Rational Roots Theorem. Let P(x) be a polynomial of degree n with integral coefficients and a nonzero constant term.

13. This leads us into what is called the Rational Roots Theorem. Let P(x) be a polynomial of degree n with integral coefficients and a nonzero constant term. P(x) = anxn + an-1xn-1 + …+ a0 where a0 ≠0

14. This leads us into what is called the Rational Roots Theorem. P(x) = anxn + an-1xn-1 + …+ a0 where a0 ≠0 If one of the roots of the equation P(x) = 0 is x = p/q where p and q are nonzero integers with no common factor other than 1, then p must be a factor of a0 and q must be a factor of an !

15. According to the rational roots theorem what are the possible rational roots of :Px) = 3x4 + 13x3 + 15x2 – 4 = 0

16. According to the rational roots theorem what are the possible rational roots of :Px) = 3x4 + 13x3 + 15x2 – 4 = 0 Note: If there are any rational roots, then they must be in the form of p/q.

17. According to the rational roots theorem what are the possible rational roots of :Px) = 3x4 + 13x3 + 15x2 – 4 = 0 Note: If there are any rational roots, then they must be in the form of p/q. 1st: List all possible q values: ±1(±3)

18. According to the rational roots theorem what are the possible rational roots of :Px) = 3x4 + 13x3 + 15x2 – 4 = 0 Note: If there are any rational roots, then they must be in the form of p/q. 1st: List all possible q values: ±1(±3) 2nd: List all possible p values: ±1(±4); (±2)(±2)

19. According to the rational roots theorem what are the possible rational roots of :Px) = 3x4 + 13x3 + 15x2 – 4 = 0 Therefore, if there is a rational root then it must come from this list of possible p/q values:

20. According to the rational roots theorem what are the possible rational roots of :Px) = 3x4 + 13x3 + 15x2 – 4 = 0 Therefore, if there is a rational root then it must come from this list of possible p/q values: p/q  ±(1/1, 1/3, 4/1, 4/3, 2/1, 2/3) which means there are 12 possibilities!

21. According to the rational roots theorem what are the possible rational roots of :Px) = 3x4 + 13x3 + 15x2 – 4 = 0 Now, determine whether any of the possible rational roots are really roots. If so, then find them.

22. According to the rational roots theorem what are the possible rational roots of :Px) = 3x4 + 13x3 + 15x2 – 4 = 0 Lets first evaluate x = 1.

23. According to the rational roots theorem what are the possible rational roots of :Px) = 3x4 + 13x3 + 15x2 – 4 = 0 Lets first evaluate x = 1. Do you remember the quick and easy way to see if x = 1 is a root?

24. According to the rational roots theorem what are the possible rational roots of :Px) = 3x4 + 13x3 + 15x2 – 4 = 0 Now, check the other possibilities using synthetic division.

25. Pg. 8425 – 39 odd Assignment: