Lesson 2-6Lesson 2-6Solving Solving

Polynomial Polynomial Equations by Equations by Factoring – Factoring –

Part 2Part 2

Objective:

To solve polynomial equations by various methods of factoring, including the use of

the rational root theorem.

Objective:

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: 3x2 – 11x – 4 = 0

Solve: 3x2 – 11x – 4 = 0

(3x + 1)(x – 4) = 0

Solving for x x = - 1/3 or x = 4

Solve:

So we only concerned ourselves with the factors of 3 and 4.

3x2 – 11x – 4 = 0(3x + 1)(x – 4) = 0

Solving for x x = - 1/3 or x = 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 theRational Roots Theorem.

This leads us into what is called theRational 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 theRational 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

This leads us into what is called theRational Roots Theorem.

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 !

P(x) = anxn + an-1xn-1 + …+ a0 where a0 ≠0

According to the rational roots theorem what are the possible

rational roots of :

Px) = 3x4 + 13x3 + 15x2 – 4 = 0

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.

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)

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)

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:

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!

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.

According to the rational roots theorem what are the possible

rational roots of :

Px) = 3x4 + 13x3 + 15x2 – 4 = 0

Lets first evaluate x = 1.

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?

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.

Pg. 8425 – 39 odd

Assignment: