lesson 2-6 solving polynomial equations by factoring – part 2
TRANSCRIPT
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: