Lesson 2 6 Solving Polynomial Equations by Factoring

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

Objective:

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

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).

Solve:

Solve: 3 x 2 – 11 x – 4 = 0

Solve: 3 x 2 – 11 x – 4 = 0 (3 x + 1)(x – 4) = 0 Solving for x x = - 1/3 or x = 4

Solve: 3 x 2 – 11 x – 4 = 0 (3 x + 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.

We call the possible factors of c p values.

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

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

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.

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-1 xn-1 + …+ a 0 where a 0 ≠ 0

This leads us into what is called the Rational Roots Theorem. P(x) = anxn + an-1 xn-1 + …+ a 0 where a 0 ≠ 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 a 0 and q must be a factor of an !

According to the rational roots theorem what are the possible rational roots of : P(x) = 3 x 4 + 13 x 3 + 15 x 2 – 4 = 0

According to the rational roots theorem what are the possible rational roots of : P(x) = 3 x 4 + 13 x 3 + 15 x 2 – 4 = 0 Note: If there any rational roots, then they must be in the form of p/q.

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

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

According to the rational roots theorem what are the possible rational roots of : P(x) = 3 x 4 + 13 x 3 + 15 x 2 – 4 = 0 Therefore, if there is a rational root then it must come from this list of possible p/q values:

According to the rational roots theorem what are the possible rational roots of : P(x) = 3 x 4 + 13 x 3 + 15 x 2 – 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!

According to the rational roots theorem what are the possible rational roots of : P(x) = 3 x 4 + 13 x 3 + 15 x 2 – 4 = 0 Now, determine whether any of the possible rational roots are really roots. If so, then find them.

According to the rational roots theorem what are the possible rational roots of : P(x) = 3 x 4 + 13 x 3 + 15 x 2 – 4 = 0 Lets first evaluate x = 1.

According to the rational roots theorem what are the possible rational roots of : P(x) = 3 x 4 + 13 x 3 + 15 x 2 – 4 = 0 Lets first evaluate x = 1. Do you remember the quick and easy way to see if x = 1 is a root?

According to the rational roots theorem what are the possible rational roots of : P(x) = 3 x 4 + 13 x 3 + 15 x 2 – 4 = 0 Now, check the other possibilities using synthetic division.

Assignment: Pg. 84 25 – 39 odd