Factoring Methods
The opposite of distribution is....
Factoring!
You get zero at the end!
Describe end behavior in your own (team's) words.
Answers vary
What are the three kinds of root behaviors?
If you take two square numbers and multiply them, the result is also a square. Why?
Ex: 4\cdot 9= 36, 25*81=2025 .
Because,
a^2\cdot b^2=(ab)^2
The following quadratic is unfactorable, explain why in detail:
x^2+8x+3
Because there are no factors of 3 that can be combined to equal 8
Simplify:
\frac(4x^3)(2x), \frac(20x^5)(4x^3), \frac(100x)(20)
2x^2, 5x^2, 5x
Why is a linear equation considered odd? Does it have odd end behavior?
What kind of exponents give each kind of root?
Cross--> exponent is 1
Bounce --> exponent is 2, or even
Wiggle --> exponent is 3, or odd
The rational root theorem says the factors of the constant divided by the factors of the leading coefficient gives you a list of POSSIBLE rational roots.
Given the following, create the list of possible rational roots:
g(x)=28x^3-179x^2+200x-25
The list includes positive and negatives of the following list:
1/1, 1/2, 1/4, 1/7, 1/14, 1/28, 5/1, 5/2, 5/4, 5/7,5/14, 5/28, 25/1, 25/2, 25/4, 25/7, 25/14, 25/28.
Factor the following:
8x^2+18x+7
(4x+7)(2x+1)
Finish the division:
\frac(20x^3+27x^2+x-6)(4x+3)
5x^2+3x-2
If I have the function P(x)=x^2+8x+12 what is the end behavior when
x\to \infty
P(x)\to \infty
given the following:
g(x)=(x-3)^2(4x-5)^7(5-x)
Find all the roots, and their behaviors
x=3, bounce. x=5/4 wiggle. x=5, cross
Graph the following:
x^3-4x^2+x+26
According to the rational root theorem, we should be able to find our roots from the list (positive and negative): 1, 2, 13, 26.
Why are we seeing only one root? Where are the other two?
They're complex! Won't see them in the xy-coordinate plane!
Factor the following:
f(x)=18x^2+102x+60
and find the roots!
6(3x+2)(x+5), the roots are: x=-5 and x=-2/3
Finish the division:
\frac(4x^5-36x^4-40x^3-7x^2+63x+70)(x^2-9x-10)
4x^3-7
An ODD degree positive polynomial tends to what value when
x\to -\infty
The odd degree positive polynomial will also tend towards -\infty .
Can a 5th degree polynomial have 3 bouncing x-intercepts?
No! that implies the exponent would be 6 instead of 5.
True or False: a quadratic can have 1 real root and one complex root.
Think really hard about how you get a complex root from a quadratic. Think about y=x^2+1 .
False! Complex roots always come in pairs. If a quadratic has one complex root, the other one is automatically complex.
Give 3 possible 'b' values so the following quadratic is factorable:
x^2+bx+48
b is made up of the sum of the factor pairs of 48. So some possible b values are 49 (48+1), 26 (24+2), 19 (3+16), 16 (4+12).
Suppose polynomial P(x) is divisible by polynomial Q(x) .
Let's suppose further that \frac(P(x))(Q(x))=L(x) . Then, compute:
L(x)\cdot Q(x)
The answer is
P(x)
Describe the full end behavior as x \to \pm \infty for
f(x)=-3x^5-9x+10
As x \to \infty, f(x)\to -\infty
As x \to -\infty, f(x) to \infty
Odd polynomials have at least 1 real root guaranteed. Explain why!
The end behavior of odd polynomials forces the polynomial to cross the x-axis.
A quadratic times a linear is what?
A cubic times a quadratic is what?
Finally, a cubic times a quadratic times a linear is what?
Q\cdot L = C
C \cdot Q = 5th degree
C\cdot Q \cdot L = 6th degree