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 Number: - 56 The numbers in the “L” are the prime factors. Factoring Variables X 3 x 3 y 2 z 3 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 1 * * * * * * * *13 (P) -1 *56 1 *56 2 *28 2 *14 2 *7 You do it: x 3 y 6 z 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 z 3 ) 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 z 3 Leftovers: 3x + 2*3 xxz - 2 or 3x + 6 x 2 z - 2) We now write ___( 4 x y z 3 ) (3x + 6 x 2 z - 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 165x 2 y x 5 y 12 (x-5) + 7x(x-5) n 3 + 3n 2 + 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) = = 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) = ( ) = - 76 negative discriminant means NO ANSWERS You can stop here. THAT IS WHY WE DO THIS FIRST!!!!!!!) Example of One answer Y= 2x 2 + 4x +2 A=2 b=4 c=2 B squared = 16 4*a*c = 4*2*2 = =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}

Vocabulary 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 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: 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. Example: 1.A = 3 B = 14 C = -5 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= x 2 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 – b 2 - 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 ax 2 + bx +c 23. Domain – x values 24.Range – y values