factoring different polynomials

8/10/2019 Factoring Different Polynomials

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FactoringPolynomials

Because polynomials come in manyshapes and sizes, there are several patterns you need to recognize and

there are different methods for solvingthem.



In Grade ten, you learned that there was a pattern withthese types of expressions. When this expression isfactored into two binomials, the two numbers will have a

product of 9 and a sum of -6.

Factoring Polynomials:

))(( x x

962 x x

Type 1:

Quadratic Trinomials with a Leading coefficient of 1

=962 x x - 3 - 3

2 Binomial Factors



Factoring Quadratic

Equations with a = 1

Try these examples:

1) x2 + 7x + 12 2) x2 + 8x + 12

3) x2 + 2x –

3 4) x2 –

6x + 8

5) x2 + x – 12 6) x2 – 3x – 10

7) x2 – 8x + 15 8) x2 – 3x – 18

9) x2 – 3x + 2 10) x2 – 10x + 21



There are a variety of ways that you can factor these typesof Trinomials:

a) Factoring by Decomposition

b) Factoring using Temporary Factors

c) Factoring using the Window Pane Method


4195 2 x x

Type 2:

Quadratic Trinomials with a Leading coefficient = 1





4195 2 x x

Type 2:


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.




Factoring Polynomials: Type 2:


1. Multiply a and c

2. Look for two numbers thatmultiply 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.

20ca

The 2 nos. are -20 & 1

4195 2 x x

41205 2 x x x

)4(1)4(5 x x x

)15)(4( x x



b) Factoring using Temporary Factors


4195 2 x x

Type 2:


1. Multiply a and c


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



20ca


4195 2 x x

)1)(20( x x

)15)(4( x x

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

5

1

5

20 x x



c) Factoring using the Window Pane Method


4195 2 x x

Type 2:


1. Multiply a and c


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



20ca


4195 2 x x

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 thatyou 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.

25 x

4 x20

x1

x5

1

x 4

)15)(4( x x



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.

9124 2 x x




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.

9124 2 x x

Perfect Squares

2

22

)3(9

)2(4

x x

And )3)(2(212 x x

9124 2 x x

)32)(32( x x



Factoring Special Cases:

Determine which of the following polynomials is a perfectsquare trinomial. If so, factor it.

1) x2 – 12x + 36

2) 9x2 + 34x + 25

3) x2 + 18x + 81

3) 64x2 - 20x + 1

Type 3: Factoring Perfect Square Trinomials

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A binomial that is created by subtracting two perfectsquares.

Ex 1: x2 – 4 Ex 2: x2 – 625

Ex 3: 4x2 – 25 Ex 4: 16x2 - 81

What is true about the factored form of each of thesebinomials?

Type 4: Factoring a Difference of Squares




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.

259 2 x




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 thefirst and second term for each

binomial.

259 2 x

Perfect Squares?

A Difference?

))(( x3 x35 5




Difference of Squares

Factor each of the following completely:

1) x2 – 100

2) x4 –

163) 100x2 – 400

4) 3x2 - 75

5) 225x2

–

121

#2 isn’t quadratic but it still can be factored!

factoring different polynomials

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