Factoring Polynomials Because polynomials come in many shapes and sizes, there are several patterns you need to recognize and there are different methods for solving them.

In Grade ten, you learned that there was a pattern with these types of expressions. When this expression is factored into two binomials, the two numbers will have a product of 9 and a sum of -6. Factoring Polynomials: Type 1: Quadratic Trinomials with a Leading coefficient of 1 = Binomial Factors

Factoring Quadratic Equations with a = 1 Try these examples: 1)x 2 + 7x + 122) x2 x2 + 8x ) x2 x2 + 2x – 34) x2 x2 – 6x + 8 5) x2 x2 + x – 126) x2 x2 – 3x – 10 7)x 2 – 8x + 158) x2 x2 – 3x – 18 9) x2 x2 – 3x + 210) x2 x2 – 10x + 21

There are a variety of ways that you can factor these types of Trinomials: a)Factoring by Decomposition b)Factoring using Temporary Factors c)Factoring using the Window Pane Method Factoring Polynomials: Type 2: Quadratic Trinomials with a Leading coefficient = 1

a)Factoring by Decomposition Factoring Polynomials: Type 2: Quadratic Trinomials with a Leading coefficient = 1 1.Multiply a and c 2.Look for two numbers that multiply to that product and add to b 3.Break down the middle term into two terms using those two numbers 4.Find the common factor for the first pair and factor it out & then find the common factor for the second pair and factor it out. 5.From the two new terms, place the common factor in one bracket and the factored out factors in the other bracket.

a)Factoring by Decomposition Factoring Polynomials: Type 2: Quadratic Trinomials with a Leading coefficient = 1 1.Multiply a and c 2.Look for two numbers that multiply to that product and add to b 3.Break down the middle term into two terms using those two numbers 4.Find the common factor for the first pair and factor it out & then find the common factor for the second pair and factor it out. 5.From the two new terms, place the common factor in one bracket and the factored out factors in the other bracket. The 2 nos. are -20 & 1

b) Factoring using Temporary Factors Factoring Polynomials: Type 2: Quadratic Trinomials with a Leading coefficient = 1 1.Multiply a and c 2.Look for two numbers that multiply to that product and add to b 3.Use those numbers as temporary factors. 4.Divide each of the number terms by a and reduce. 5.Multiply one bracket by its denominator

a)Factoring by Temporary Factors Factoring Polynomials: Type 2: Quadratic Trinomials with a Leading coefficient = 1 The 2 nos. are -20 & 1 1. Multiply a and c 2. Look for two numbers that multiply to that product and add to b 3.Use those numbers as temporary factors. 4.Divide each of the number terms by a and reduce. 5.Multiply one bracket by its denominator

c) Factoring using the Window Pane Method Factoring Polynomials: Type 2: Quadratic Trinomials with a Leading coefficient = 1 1.Multiply a and c 2.Look for two numbers that multiply to that product and add to b 3.Draw a Windowpane with four panes. Put the first term in the top left pane and the third term in the bottom right pane. 4.Use the two numbers for two x-terms that you put in the other two panes 5.Take the common factor out of each row using the sign of the first pane. Take the common factor out of each column using the sign of the top pane. These are your factors.

a)Factoring by Temporary Factors Factoring Polynomials: Type 2: Quadratic Trinomials with a Leading coefficient = 1 The 2 nos. are -20 & 1 1. Multiply a and c Look for two numbers that multiply to that product and add to b 2. Draw a Windowpane with four panes. Put the first term in the top left pane and the third term in the bottom right pane. 3. Use the two numbers for two x-terms that you put in the other two panes. 4.Take the common factor out of each row using the sign of the first pane. Take the common factor out of each column using the sign of the top pane. These are your factors.

Special Case Trinomials: Loooking for Patterns: A trinomial formed by squaring a binomial. Ex1: (x + 5) 2 Ex2: (2x – 3) 2 Ex3: (x - 4) 2 Ex4: (5x + 2) 2 What do you notice about the resulting trinomials?

Factoring Special Cases: Type 3: Factoring Perfect Square Trinomials 1.First determine if the first and third terms are perfect squares. Identify their square roots. 2. Determine if the middle term is twice the product of those square roots. If so, then this Trinomial is a Perfect Square Trinomial! 3. Set up two brackets putting the square roots in as the first and second term for each binomial.

Factoring Special Cases: Type 3: Factoring Perfect Square Trinomials 1.First determine if the first and third terms are perfect squares. Identify their square roots. 2. Determine if the middle term is twice the product of those square roots. If so, then this Trinomial is a Perfect Square Trinomial! 3. Set up two brackets putting the square roots in as the first and second term for each binomial. Perfect Squares And

Factoring Special Cases: Determine which of the following polynomials is a perfect square trinomial. If so, factor it. 1)x 2 – 12x )9x x )x x )64x x + 1 Type 3: Factoring Perfect Square Trinomials

Factoring Special Cases: A binomial that is created by subtracting two perfect squares. Ex 1: x 2 – 4Ex 2: x2 x2 – 625 Ex 3: 4x 2 – 25Ex 4: 16x What is true about the factored form of each of these binomials? Type 4: Factoring a Difference of Squares

Factoring Special Cases: Type 4: Factoring A Difference of Squares 1.First determine if the two terms are perfect squares. Identify their square roots. 2.Set up two brackets, one with an addition sign and the other with a subtraction sign. They are different, get it? 3.Then insert the square roots in as the first and second term for each binomial.

Factoring Special Cases: Type 4: Factoring A Difference of Squares 1.First determine if the two terms are perfect squares. Identify their square roots. 2.Set up two brackets, one with an addition sign and the other with a subtraction sign. They are different, get it? 3.Then insert the square roots in as the first and second term for each binomial. Perfect Squares? A Difference?

Factoring Special Cases: Difference of Squares Factor each of the following completely: 1) x2 x2 – 100 2) x 4 – 16 3) 100x 2 – 400 4) 3x ) 225x 2 – 121 #2 isn’t quadratic but it still can be factored!