Bell Ringer  1. What is a factor tree?  2. What are the terms at the bottom of a factor tree called?  3. What is GCF?

Factoring Polynomials Thursday October 2, 2014

Greatest Common Factor  No matter what type of polynomial you are factoring, you always factor out the GCF first!

What if it’s a binomial?  1 st – Factor out GCF  2 nd – Difference of Squares  3 rd – Sum of Cubes  4 th – Difference of Cubes

Binomials continued …  Difference of squares – Ex: (4x 2 – 9)  (2x + 3) (2x – 3)  Sum of cubes – Ex: 8x  (2x +3) (4x 2 – 6x + 9)  Difference of cubes – Ex: x 3 – 8  (x – 2) (x 2 + 2x + 4)

What if it’s a trinomial?  1 st – Factor out GCF  2 nd – Perfect Square Trinomial  3 rd – “Unfoil”

Trinomials continued…  1 st term is a perfect square, last term is a perfect square, middle term is double the product of the square roots of the first and last terms. Then, subtract or add depending on sign of middle term.  Ex: 4x 2 – 4x +1  (2x -1) 2 Square root of 4x 2 is 2x, square root of 1 is 1, 2(2x * 1) = 4x  Ex: 9x x + 16  (3x + 4) 2 Square root of 9x 2 is 3x, square root of 16 is 4, 2(3x * 4) = 24x

Trinomials continued… “Unfoil”  Find the factors of the first and last terms. How can we get the middle term with them?  If it’s a + and + or a – and +, you need to multiply and then add to get the middle term. You will factor as a - - or a + +.  If it’s a + and -, then you need to multiply then subtract to get the middle term. You will factor as a + -.

Examples:  If it’s a + and + or a – and +, you need to multiply and then add to get the middle term. You will factor as a + + or a - -.  a 2 + 7a + 6 = (a + 6) (a + 1)  x 2 – 5x + 6 = (x – 3) (x – 2)

Examples:  If it’s a + and -, then you need to multiply then subtract to get the middle term. You will factor as a + -.  x 2 + 4x – 5 = (x + 5) (x – 1)

Uncover the mystery of factoring complex trinomials!

Tic-Tac-But No Toe Part 1: In the following tic tac’s there are four numbers. Find the relationship that the two numbers on the right have with the two numbers on the left. 1. What did you find? 2. 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 Tac’s 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. ax 2 + 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– WHERE’S 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 ½ and -3/5, your answer would be (x + 2) (3x – 5).

EXAMPLES X 2 – X ? ? 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 2X 2 + 8X ? 4 ? What 2 numbers complete the Tic Tac? Since a = 1, put a 1 in for the numerator in two fractions. 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). *Remember that sometimes a GCF should be factored out before beginning. 2(X 2 + 4X – 32)

EXAMPLES 1/2X 2 + 1/2X ? 1 ? 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 ½ (x – 3) (x + 4). *Remember that sometimes a GCF should be factored out before beginning. 1/2(X 2 + X – 12)