Factoring…Taking Polynomials apart Name____________________________ Period________

Prime FactoringWhat are Prime numbers?_______________________List the prime number starting with 1 ____________________________________________The “L” method of factoring. Number 300300

Number: - 56

The numbers in the “L” are the prime factors .Factoring VariablesX 3 x3 y2 z3 12 x 2 y z 3

X x y z 1*12 x y zX x y z 2*6 X zX x z 2 *3 z

1 * 3003002 *

150150

2 * 750753 * 250255 * 50055 * 10017 * 14311 * 13 (P)

-1 * 561 *

56

2 * 282 *

14

2 * 7

You do it:-9282 945 320

36 x 3 y 6 z 3 5022 x 6 y z 2 m

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: • y = Ax 2 + Bx + C ( Called Standard Form)On the Quadratic below, A=6 B =-2 and C=4• 6x 2 – 2x - 4• First we find the discriminant to find out how many

answers there will be (two, one or none)• 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????

•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 ANSWERS You can stop here. THAT IS WHY WE DO THIS FIRST!!!!!!!)

Example of One answer

Y= 2x2 + 4x +2A=2 b=4 c=2

B squared = 164*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}

Vocabulary1. Quadratic Parent equation 2. Parabola3. Prime Number4. Factor5. Greatest Common Factor6. term7. Polynomial8. Monomial9. Binomial10. Trinomial11. Quadratic Formula12. Discriminate13. Axis of Symmetry13. Axis of Symmetry Formula14. Reflection15. Vertex16. Maximum17.Minimum

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:

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: AnswersNone is the discriminate is negativeOne if the discriminant is zero Two if the discriminant is positiveSo, 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.

Example:

1.A = 3 B = 14 C = -5

Let’s checkFactor

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 quadratic3. Prime Number- A number that can only be divided evenly by itself and 14. Factor- are numbers or terms you can multiply together to get another number or term5. Greatest Common Factor- The largest number (or term) that two or more number (or terms) have in common6. Term – A string of numbers (and variables) connected by multiplication7. Polynomial- A string of terms connected by addition or subtraction. 8. Monomial- A polynomial with one term9. Binomial- A polynomial with two terms10. Trinomial- A polynomial with three terms11. Discriminate – b2 - 4ac it determines if a quadratic has one, two or no answers12. Zeroes – solutions to a quadratic13. 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, solutions15, 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 graph17. Axis of Symmetry Formula -b/2a (the opposite of b divided by 2 times a18. Reflection – The mirror image of a point, in quadratics mirrored over the axis of symmetry19. 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 +c23. Domain – x values24.Range – y values