Introduction to Complex Numbers

Introduction to Complex Numbers

Bell Ringer 2-23-17 1. What is a factor tree? 2. What are the terms at the bottom of a factor tree called? 3. What is GCF? 4. What does prime mean? Factoring Polynomials Completely Thursday February 23, 2017 Thursday September 29, 2016 Greatest Common Factor No matter what type of polynomial you are factoring, you always factor out the GCF first! What if its a binomial?

1st Factor out GCF 2nd Difference of Squares 3rd Sum of Cubes 4th Difference of Cubes Binomials continued Difference of squares Ex: (4x2 9) (2x + 3) (2x 3) Sum of cubes Ex: 8x3 + 27 (2x +3) (4x2 6x + 9) Difference of cubes Ex: x3 8 (x 2) (x2 + 2x + 4) What if its a trinomial? 1st Factor out GCF 2nd Perfect Square Trinomial 3rd Unfoil or Unbox

Factoring Fanatic Uncover the mystery of factoring complex trinomials! Tic-Tac-But No Toe Part 1: In the following tic tacs there are four numbers. Find the relationship that the two numbers on the right have with the two numbers on the left. -90 10 36 -6 -36 -6

-30 -6 1 -9 -12 -6 0 6 -1 5 -49 7

120 30 -81 9 -24 -6 0 -7 34

4 0 -9 -10 -4 -72 24 16 4 -6 -3 49

-7 21 -3 8 4 -1 -14 -7 Observations

1. 2. 2 What did you find? Did it follow the pattern every time? Tic-Tac-But No Toe Part 2: Use your discoveries from Part 1 to complete the following Tic Tacs. 9 16 18 6

-35 10 -10 9 7 2 4 45 6 -3

-15 -5 14 -5 -2 2 72 -6 -72

-36 -22 -38 -5 -1 5 9 Observations 3. Did your discovery work in every case?

4. Can you give any explanation for this? Finally! Factoring with a Frenzy! Arrange the expression in descending (or ascending) order. ax2 + bx + c = 0 Be sure the leading coefficient is positive. Factor out the GCF, if necessary. Multiply the coefficients a and c and put the result in quadrant II of the Tic Tac. Put the coefficient b in quadrant III of the Tic Tac. Play the game! Just like the previous problems. (Find the relationship!) Once you have completed your Tic Tac WHERES the ANSWER?

Use the a coefficient as the numerator of two fractions. Use the results in quadrants I and IV as the two denominators. Reduce the fractions. The numerator is your coefficient for x in your binominal and the denominator is the constant term. EXAMPLE: If you get the fractions and -3/5, your answer would be (x + 2) (3x 5). EXAMPLES X2 X - 12 -12 ? -1 -12 3 -1 -4

? What 2 numbers complete the Tic Tac? Since a = 1, put a 1 in for the numerator in two fractions. You found 3 and -4. These are the denominators for the two fractions. Your fractions are 1/3 and 1/4 Your answer is (x + 3) (x 4). EXAMPLES 2X2 + 8X - 64 *Remember that -32 ?

What 2 numbers sometimes a GCF complete the Tic Tac? should be factored 4 ? out before beginning. 2(X2 + 4X 32) Since a = 1, put a 1 in for the numerator in two fractions. -32 8 4 -4 You found 8 and -4. These are the denominators for the two fractions. Your fractions are 1/8 and 1/4.

Your answer is 2 (x + 8) (x 4). EXAMPLES 1/2X2 + 1/2X - 6 *Remember that -12 ? What 2 numbers sometimes a GCF complete the Tic Tac? should be factored 1 ? out before beginning. 1/2(X2 + X 12) Since a = 1, put a 1 in for the -12 -3 numerator in two fractions.

1 4 You found -3 and 4. These are the denominators for the two fractions. Your fractions are 1/3 and 1/4. Your answer is (x 3) (x + 4). EXAMPLES 3X2 + 5X = 12 *Remember to re-write in standard form 3X2 + 5X - 12 -36 9

5 -4 -36 ? 5 ? What 2 numbers complete the Tic Tac? Since a = 3, put a 3 in for the numerator in two fractions. You found 9 and -4. These are the denominators for the two fractions. Your fractions are 3/9 = 1/3 and 3/4 Your answer is (x + 3) (3x 4).

What if its a polynomial of 4 or more? 1st Factor out GCF 2nd Factor by Grouping Factoring by Grouping Ex: x3 + 3x2 + 2x +6 1.Group two terms together. (x3 + 3x2) + (2x + 6) 2. Factor out a GCF from each separate binomial to get a common binomial. x2 (x + 3) + 2(x + 3) 3. Factor out the common binomial. (x+3) (x2 + 2) Factoring Completely Remember, GCF first always. Check for your Special Cases. Sometimes you may have to

factor more than once. Sometimes it will be Prime. Assignments Classwork: Factoring Polynomials #1-20 Homework: Factoring Polynomials #21-40 Exit Ticket 1. What do you ALWAYS do first when factoring? 2. What does it mean if the polynomial is prime? 3. How many factors can a polynomial have?

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