Transcript Factoring…Taking Polynomials apart

Factoring…Taking Polynomials apart Name____________________________ Period________
Prime Factoring
What are Prime numbers?_______________________
List the prime number starting with 1
____________________________________________
The “L” method of factoring. Number 300300
1
* 300300
2
* 150150
-1 *
56
2
* 75075
1 *
56
3
* 25025
2 *
28
5
* 5005
2 *
14
5
* 1001
2 *
7
7
* 143
You do it:
-9282
945
320
Number: - 56
36 x 3 y 6 z 3
5022 x 6 y z 2 m
11 * 13 (P)
The numbers in the “L” are the prime factors .
Factoring Variables
X3
x3 y2 z3
12 x 2 y z 3
X
x y z
1*12 x y z
X
x y z
2*6 X z
X
x
z
2 *3
z
Try:
(12 x 2 y z 3 )
+
(24 x 3 y z 5) - (8y)
Lunch and leftovers Factoring Polynomials
•
•
•
•
•
•
•
•
•
•
•
•
The last problem was a lead in to factoring
polynomials. When you factor a polynomial, you
use the L method to factor each term, then
gather up all the common factors for lunch and
leave the leftovers. (we will presume the 1
factor)
12 x 2 y z 3 ) +
(24 x 3 y z 4) - (8 x y z3 )
2*6 x y z
2*12 x y z - 2*4 x y z
2*3 x
z
2 *6 x z
2*2
z
z
2*3 x z
z
z
Gather up the common factors for lunch (Cross
out as we do)
Lunch = 2*2 x y zzz or 4 x y z3
Leftovers: 3x + 2*3 xxz - 2 or 3x + 6 x2z - 2)
We now write ___( 4 x y z3) (3x + 6 x2z - 2)
Lunch (Leftovers)
Factoring is really (Un distributing)
Greatest Common Factor. The “lunches”are
the greatest common factors of the terms.
Greatest common factor is the biggest
number or variable power that “goes into”
each term evenly.
You do it:
GCF 28 and 44
GCF
165x2 y3
429 x 5 y
12 (x-5) + 7x(x-5)
n3 + 3n2 + 4n + 12 (split method)
SOLVING Quadratics
•
There are many ways to factor quadratics but we
will only concentrate on three
1. Factoring and 2. the quadratic formula. 3. Graphing
2. The quadratic formula works ALL THE TIME
• Factoring
• First we have to know the parts of the Quadratic:
•Example of No answers!!!!
• 5x 2 – 2x +4 Step 1: A = 5 B= -2 C
=4
•Step 2: Discriminant
(B 2 – 4AC) = (-2)(-2) - (4)(5)(4) =
( 4 - 80 ) = - 76
negative discriminant means NO
•
y = Ax 2 + Bx + C ( Called Standard Form)
ANSWERS
On the Quadratic below, A=6 B =-2 and C=4
You can stop here. THAT IS WHY WE
•
6x 2 – 2x - 4
• First we find the discriminant to find out how many DO THIS FIRST!!!!!!!)
answers there will be (two, one or none)
Example of One answer
•
•
•
1. A positive discriminant = two answers
2. A negative discriminate means no answers
3,. A discriminant of zero = one answer
• The DISCRIMINANT = ( b 2 - 4ac)
•
So, in this problem we have (-2)2 -4(6)(-4) or 4 – (96) = 4 +96 = 100 So, there will be TWO answers.
•
THIS IS IMPORTANT AS IF THERE ARE NO
ANSWERS, WHY CONTINUE????
Y= 2x2 + 4x +2
A=2 b=4 c=2
B squared = 16
4*a*c = 4*2*2 = 16
16-16 =0 means one answer
Homework show all work On separate paper.
Why do we do this? To find the “solutions” to a
quadratic equation, you just set your factors = to zero
and solve. Those are your solutions, roots, zeros,
answers. Example if you got the two factors(2x -3) (4x
+5), to find the solutions:
2x -3 = 0 x = 3/2 (1.5) 4x +5= 0 x = -5/4 (-1.25)
Written as the solution SET { 1.5, -1.25} or { 3/2, -5/4}
1. Quadratic Parent equation
2. Parabola
3. Prime Number
4. Factor
5. Greatest Common Factor
6. term
7. Polynomial
8. Monomial
9. Binomial
10. Trinomial
11. Quadratic Formula
12. Discriminate
13. Axis of Symmetry
13. Axis of Symmetry Formula
14. Reflection
15. Vertex
16. Maximum
17.Minimum
Vocabulary
18. Zeros
19. Roots
20. X Intercept
21. Y Intercept
22. Solutions
23. Domain
24. Range
25. Regression
The Quadratic Formula
You can always find the solutions to any
quadratic equation using the quadratic
formula:
Example:
1. A = 3 B = 14 C = -5
Look under the square root sign…it’s the
discriminant!!!!!!
That is why a negative discriminant has no
answer…you cannot take the square root of a
negative number in the real number world!!!!
This is just a game of alphabet soup. You find
you’re A, B and C (same in big case as little
case). Plug in the number and: Answers
None is the discriminate is negative
One if the discriminant is zero
Two if the discriminant is positive
So, always do the discriminant first. If it is
negative, why do all that work!!!! And you will
know if you got the correct number of
answers.
Let’s check
Factor
Name: __________________period____________________
Now finish solving (if possible) using the quadratic formula and write the solutions in set form
and in factor form.
And put in set form and factor form
1. Quadratic Parent equation - y= x2
2. Parabola –A U shape made by graphing a
quadratic
3. Prime Number- A number that can only
be divided evenly by itself and 1
4. Factor- are numbers or terms you can
multiply together to get another number or
term
5. Greatest Common Factor- The largest
number (or term) that two or more number
(or terms) have in common
6. Term – A string of numbers (and variables)
connected by multiplication
7. Polynomial- A string of terms connected
by addition or subtraction.
8. Monomial- A polynomial with one term
9. Binomial- A polynomial with two terms
10. Trinomial- A polynomial with three
terms
11. Discriminate – b2 - 4ac it determines if a
quadratic has one, two or no answers
12. Zeroes – solutions to a quadratic
13. Roots– solutions to a quadratic
25. Quadratic Formula
14. X intercept – where a graph crosses or
touches the x axis In a quadratic, the roots,
zeroes, solutions
15, Y intercept – where a graph touches or
crosses the y axis in a quadratic, it is at “c”.
16. Axis of Symmetry – the vertical line that
passes through the vertex of a quadratic graph
17. Axis of Symmetry Formula -b/2a (the
opposite of b divided by 2 times a
18. Reflection – The mirror image of a point, in
quadratics mirrored over the axis of symmetry
19. Vertex – the highest or lowest point on a
quadratic graph found by using the axis of
symmetry as “x’ in the quadratic equation
given.
20. Maximum – a vertex of a parabola that
opens down (a is negative)
21. Minimum – a vertex of a parabola that
opens up (a is positive)
22. Standard Form Quadratic ax2 + bx +c
23. Domain – x values
24. Range – y values