262 xx 212 xx 234 xxxxx 322

Any combination of the prime factorization.

)Find the number that “GAZINTA” all the numbers.

62 3

6 goes into 12, 2 times and into 18, 3 times.

The only number that divides into 2 and 3 is 1…the number on the left side is the GCF.GCF = 6

)46 8)23 4

The product of the numbers on the left side is the GCF.GCF = 4 * 2 = 8

)44 6 9

There are no common factors in the remaining factors of 4, 6, and 9. The GCF = 4

The 6 and the 8 can still be divided by 2.

From the previous examples.

3626 x

326 xLeftovers

ba 4838

ba 438 Leftovers

946444 2 xx

9644 2 xxLeftovers

Distributive Property.

1343 x 143 x

babaab 1044

When variables are a GCF, it will always be to the smallest power.

baab 104 26x 3x x4 3

223 yx 3x xy5 32xy 23y yx a2 b3

Group the terms in half.

Factor each side by GCF’s _________ _________ 2x x5 1

Since the 1st two terms are subtracting, both ( )’s will have minus signs.

3 x5 1

In fact, both sets of ( )’s must be the same for this to factor. If not the same, prime.Factor the same binomials as a GCF.

____15 x 2x 3

Factor by grouping

482 23 xxx 212868 34 xxx

byaybxax 2436 2135106 xwxw

_________ _________ 22x x 4 1 x 4

____4x 22x 1

_________ _________ x2 34x 3 7 34x 3

____34 3 x x2 7

_________ _________ x3 a2 b y2 a2 b

____2 ba x3 y2

_________ _________ w2 3 x5 7 3 x5

____35 x w2 7

There is another pattern. Find the product of the F & L terms and O & I terms.

Make up any two binomials, with no GCF, and FOIL them.

F.O.I.L.

When the “c” term is positive it means that the binomials have the same signs, and the sign on the “b” term determines the signs of the binomials.

When the “c” term is negative it means that the binomials have the opposite signs, and the sign on the “b” term determines the signs of the largest value in the binomials.

210xLF 210xIO

210xLF 210xIO

210xLF 210xIO

210xLF 210xIO

This pattern gives us the a, b, c rule for finding our factors. Number Sense!

Factor the trinomials.

1282 xx 2082 xx 652 xx

cbxax 2

b

ca

______

______

O Ix xcIxOxax 2

bx

ca

______

______

_________ _________

O I

Factor by Grouping

8______

121______

b

ca

________

Answer looks like.

________

6 26 2

12262 xxx

_________ _________ 26x x 6x

6x 2x

8______

201______

b

ca

________ BigAnswer looks like.

10 2

10 2

202102 xxx

_________ _________ 210x x 10x

2x10x

Did you just notice that the numbers in the binomial answers are the same numbers that were our factors?

This will always happen when a = 1!

Answer looks like.

5______

61______

b

ca

2323

3x 2x

Number Sense Rules.Odd + Even = OddEven + Even = EvenOdd + Odd = Even2 is a factor every Even.(Odd)*(Odd) = Odd

Factor the trinomials.

____ xx 6 1

5______

61______

b

ca6 16 1

____ xx 3 7

4______

211______

b

ca3 73 7

____ xx 4 6

10______

241______

b

ca4 64 6

____ xx 2 12

___ ___ 24

___ ___ 10

c

b

2 122 12

____ xx 3 8

___ ___ 24

___ ___ 5

c

b

3 83 8

____ xx 3 8

___ ___ 24

___ ___ 11

c

b

3 83 8

____ xx 5 15

___ ___ 75

___ ___ 20

c

b

5 155 15

____ xx 4 12

___ ___ 48

___ ___ 8

c

b

4 124 12

____ xx

___ ___ 60

___ ___ 7

c

b

5 125 12

5(-12) = -60 not 60!

Prime

even + odd

odd

odd + odd

even

(odd)*(odd) = odd

even + even

even

(odd)(even)=even

even + even

even

even

odd + even

odd

odd + even

odd

odd + odd

even

odd

even + even

even

even

Prime doesn’t happen too often, so make sure you check everything!

____ xx 2 18

___ ___ 36

___ ___ 20

c

b

2 182 18

____ xx 4 5

___ ___ 20

___ ___ 1

c

b

4 54 5

____ xx 1 6

___ ___ 6

___ ___ 7

c

b

1 61 6

____ xx

___ ___ 8

___ ___ 7

c

b

1 81 8

1(-8) = -8 not 8!

Prime yxyx ____ 15 3

___ ___ 45

___ ___ 12

c

b

15 3

15 3

nmnm ____ 3 8

___ ___ 24

___ ___ 5

c

b

3 83 8

even + even

even

even

odd + odd

even

odd

These directions means more than one factoring…Watch for GCF!

65 2 xxGCF of 5

____5 xx 2 3

___ ___ 6

___ ___ 1

c

b

2 32 3

13222 xxx

GCF of x2

____2 xxx 12 11

___ ___ 132

___ ___ 1

c

b

12 1112 11

543 26 xxx

GCF of -3x6

____3 6 xxx 1 5

___ ___ 5

___ ___ 4

c

b

1 51 5

1862 2 xxGCF of -2

____2 xx

___ ___ 18

___ ___ 6

c

b

? ?

? ?

1862 2 xx

Not Prime…factored -2 out!

3223 1224 yyxx

GCF of 4

Can’t go any further because of the variables cubed.even + even

even

even

F.O.I.L.

When the “c” term is positive it means that the binomials have the same signs, and the sign on the “b” term determines the signs of the binomials.

When the “c” term is negative it means that the binomials have the opposite signs, and the sign on the “b” term determines the signs of the largest value in the FACTORS not the binomials.

260xLF 260xIO

260xLF 260xIO

260xLF 260xIO

260xLF 260xIO

The author actually suggested guessing what the binomials are and FOILing them out to test if the middle term is correct. 8 tries to get the right answer!?!

7______

1210______

b

ca

________

Answer looks like.

1281510 2 xxx

_________ _________ 43x5 x2 3x2

Refers to the middle term. ODD + EVEN = ODD

odd even

Since we have an odd + even, we need odd factors. Break the 10 and 12 down to odd factors.

4253

5 432Isolate the odd factors and multiply all possible odd combinations.

15 8___3 ___5 ___15

7403 Not the factors

4235 2440

7245 Not the factors

4235 8

7815 Right factors

15 8

3x2 4x5

It should still factor if we switch the 15x and -8x.

________

1215810 2 xxx

_________ _________ 34x2 x5 4x5

4x5 3x2

I can see a pattern! When you look at the left side of each factoring by grouping, I see the two binomials in the answer!Do you see that? Say YES! What terms are generating these binomials? Look above each step.

It is the leading term and the two factors! Can we all agree that we will always factor out at least an x as the GCF? Yep.

Here is a shortcut. Always put the “a” in both binomials.

__10__10 xxPut in the factors.

158

Take out GCF’s

2

45 x

5

32 x

Refers to the middle term. EVEN + EVEN = EVEN

__8__8 xx

Answer looks like this using the new short cut. Use 8x twice.

14__________

158__________

b

ca

even evenBecause a = 8

Since we have an even + even, we factor out a 2 from our factors. Break the 8 down to get factors of 2’s. Put a 2( ) in each blank as a factor because we know that the two factors are even.

53222

22( ) 2( )Factor 2 out of the -14. The sum of the two red ( )’s must = (– 7).

Since – 7 is odd. Isolate the odd factors and multiply all possible odd combinations.

___3 523

10

7103 Right factors! Put them in the red ( )’s!

___5 ___15

3 10

6 20

6 and -20 are the two factors that add up to -14. Place them in our answer.

6 20 Now we know we are not finished because we used the 8 twice. We have to divide out the extra 8 by finding the GCF of each binomial.2

34 x

4

52 x

( – 7 )

EVEN RULE ( odd + odd )

Factor. 63343 2 xx

Refers to the middle term. ODD + ODD = EVEN

__3__3 xx

Answer looks like this using new short cut. Use 3x twice.

34__________

633__________

b

ca

odd oddBecause both a & c are odd

Since a and c as odd factors we have an odd + odd = -34. This is going to take some time because all the factors will be odd. Break 63 down.

733

Isolate each odd factor, from smallest to largest, and then multiply all possible odd combinations to create more odd factors.

___3 ___7 ___9 ___21 ___27 7333 63

34633 Wrong factors!

3337 27

34277 Right factors!

7 27

7 27

-7 and -27 are the two factors that add up to -34. Place them in our answer.

7 27 Now we know we are not finished because we used the 3 twice. We have to divide out the extra 3 by finding the GCF of each binomial. 73 x

3

9x

NO GCF

These directions means more than one factoring…Watch for GCF!

__3__3 xx

10__________

83__________

b

ca

even evenBecause c = -8

42 The 2 or 4 must be multiplied to the 32 12

2 12

2 12 23 x3

4xNO

GCF Remember this example 2 pages ago, where the author FOILed it out 8 times? Which way is easier?

__5__5 xx

7__________

65__________

b

ca

odd even

The 5 and -6 doesn’t work, so try 3 and 10!-3 and 10 work.

3 103 10

10 3 2x5

35 xNO

GCF

32

GCF of 2x. 31142 2 xxx

__4__42 xxx

11__________

34__________

b

ca

Because 11 is much bigger than 4 and 3, multiply 4 and 3 to get 12.

1 121 12

112 32 xx4

14 xNO GCF

__4__4 xx

12__________

54__________

b

ca

odd even

even evenBecause a = 4

2 2

One of the 2’s must be multiplied to the 5. 2 and 10

2 102 10

2 10 12 x2 2

52 x

Factor completely.

xyyx 1946 22 186310 2 xx

20236 2 xx 968 2 xx

22 4196 yxyx

Need to have x powers in descending order.

yxyx __6__6

19__________

46__________

b

ca1 243 8

odd even odd

32

No possible factors.

PRIME __10__10 xx

63__________

1810__________

b

ca

332

3 603 60

3 60 310 x 6x10NO

GCF

odd even odd

52

63 is a big value… factors must be far apart.

__6__6 xx

23__________

206__________

b

ca

522

15 8

15 8

15 8 52 x 43 x3

odd even odd

32

3*_____5*_____15*____

40

24

8

2

__8__8 xx

6__________

98__________

b

ca

222

6 12

3 6

12 6 32 x 34 x4

even even even

33

2( ) 2( )

2

32

One of the 3’s must be mult. to the 2, 6 and 3 subtract to be the 3 in the ( )’s.

3*_____5*_____9* ____15*____45*____

60

Factor completely.234 302830 xxx 183918 2 xx

22 28176 yxyx 152812 2 xx

GCF of 2x2. 1514152 22 xxx

__15__152 2 xxx

14__________

1515__________

b

ca

odd odd even

53 53

3*_____5*_____9*____

75

45

25

1514152 22 xxx

GCF of -3.

61363 2 xx

__6__63 xx

13__________

66__________

b

ca

23 23

odd even odd

9 49 4

9 4

323 x 23 x3 2

yxyx __6__6

17__________

286__________

b

ca

odd even odd

72232

3*_____7*_____21*____

56

24

8

7 247 24

24 7NO

GCF

yx 76 yx 4

6

__12__12 xx

28__________

1512__________

b

ca

322

18 109 5

18 10

32 x 56 x

6

even even even

53

2( ) 2( )

2

142

The two 3’s mult. together, 9 and 5 add up to be the -14 in the ( )’s.

It is important to know that x2, x4, xeven, etc. are all perfect squares

442 xxPer.Sqr.

Per.Sqr.

Let’s try a ( )2

22xWe must test the middle term!

xxx 422 It Factors!

25x

xxx 1055

Done!

24x

xxx 844

Done!

23x

xxx 633

NOT Done!

ERASE

____ xx 1 9Done!

4 and 36 are perfect squares, but 4 is a GCF!

964 2 xx

234 x

xxx 633

Done!

234 x

xxx 241212

Done!

273 x

xxx 422121

Done!

HIDE inside other polynomials!

55 xx 88 xx 49 2 x

GCF of 9, 1st!

229 xx

Factor completely.

8116 4 x 42 x 7512 2 x

22 2536 yx 2864 121100 yxca

9494 22 xx

323294 2 xxxAnother Diff. of Per. SQ!

NOT Diff. of Per. SQ! PRIME

GCF of 3, 1st!

2543 2 x

52523 xx

22 3625 xy

22 25361 yx GCF of -1, 1st!

yxyx 56561

Or put the 25y2 in front.

xyxy 6565

yxcayxca 432432 11101110

Even powers on the variables are still perfect squares. Divide the powers by 2 to take the square root.

Middle terms exist!

Binomial Trinomial

_______________33 ba

Same sign as given

Opposite sign as given

Always Plus

3 3 a b

2a

ab 2b

_________________

3 3

y3 x2

29y

xy6 24x __________________

3 3

x4 5

216x

x20 25

_________________

3 3

yx2 6

24 yx

yx26 36 __________________

3 3

x10 7

2100x

x70 49

3382 txa

______________2 a

3 3

x2 t

24x

xt2 2t

Both Diff. of Per. Sq. and Perfect Cubes

1st

2nd

1818 33 xx

____________

3 3

x2 1

24x

x2 1 ____________ x2 1 24x x2 1

GCF( LEFTOVERS )The number of terms in the leftovers determines which step we go to.

bababa 221st

2nd

Remember these like to hide inside of other polynomials.

_______________33 ba

Same sign as given

Opposite sign as given

Always Plus

3 3 a b

2a

ab 2b

Remember the sign rules for what your answer looks like.

____ axaxb

ca

__________

__________d e

d ed e

GCF GCF

_________ _________

dcxbxax 23

GCFLT

GCFRT

SAME SAME

____________ SAME

GCFLT

GCFRT

2222 2 ybabxxa

3 to 1 SPLIT

22 ybax Difference of

Perfect Squares

1 to 3 SPLITis the same concept, but watch for signs!

GCF of 5 165 4 x

Step 2D.P.S. 445 22 xx

Step 2D.P.S.Again

2245 2 xxx

GCF of 2x 4518522 23 xxxx

Step 4F. by G. 529522 2 xxxx

9522 2 xxxStep 2D.P.S. 33522 xxxx

GCF of 3 883 235 xxx

Step 4F. by G. 1813 223 xxx

813 32 xxStep 2D.P.S.& P.C.

422113 2 xxxxx

Step 2D.P.S. 88 33 xx

Step 2& P.C. twice 422422 22 xxxxxx

Factor completely.

GCF of 7 657 2 xx

Step 3 237 xx

GCF of 3x2 24103 22 xxx

Step 3 2123 2 xxx

3 to 1 SPLIT

2 23x y

Step 4F. by G.

yxyx 33

Difference of Per. Squares

yxyx 33

1 to 3 SPLIT

22 3 xy

Step 4F. by G.

33 xyxy

Difference of Per. Squares

33 xyxy

9622 xxyGCF of -1 first.

Distribute the minus!

Remember the product of -8(10)(3)(5)(0)(7)(11) = 0.

05 x 02 xSolve each for x.

5 5

0x

22

2x2,0x

07 08 x 032 x07 8x 32 x

2

3x

2

3,8x

02 x0x

01x1x

01x1x

1,1,0 x

064 xx04 x 06 x

4x 6x6,4x

0212 xx

12x 2x

2,12 x

Solve the equations by factoring.

01682 xx xx 62 254 2 x

044 xx

4xWe don’t have to list the same number twice, but just know that there were two answers that were the same value.

Never divide by the variable!Set the equation = 0.

062 xx 06 xx

0x 6x

6,0x

0254 2 x

05252 xx

052 x

52 x

2

5x

2

5x

2

5,

2

5x

No reason to work out the 2nd binomial because the only difference will be the sign.

Solve the equations by factoring.

063162 xx 28122 xx

02482 23 xxx 0356 2 xx

097 xx

7x 9x

9,7 x

028122 xx 0214 xx

14x 2x2,14x

01242 2 xxx

0622 xxx

6x2x0x

6,2,0 x

0__6__6 xx

1__________

356__________

b

ca

57

15 14

15 14

1514 73 x 052 x2

odd even odd

32

3

The factors have to

differ by 1, so 2(7)=14

and 3(5)=15

073 x73 x

3

7x

052 x52 x

2

5x

Solve the equations by factoring.

027308 2 xx012832 23 xxx

0__8__8 xx

30__________

278__________

b

ca

222

36 6

18 3

36 6

even even even

333

2( ) 2( ) 152

One of the 3’s must be isolated, 3 and 18 will subtract to be 15 in the ( )’s.

092 x

92 x

2

9x

034 x

34 x

4

3x

0324322 xxx

0432 2 xx

02232 xxx

032 x

32 x

2

3x

2x 2x

4

3,

2

9x

2

3,2,2 x

4 2

Solve the equations by factoring.

9123 xx xxx 543 Will need to FOIL and set = 0

9362 2 xxx01252 2 xx

0__2__2 xx

5__________

122__________

b

ca

322

odd even odd

33 8

8

8 3

04 x

4x

032 x

32 x

2

3x

4,2

3x

xxxx 512342

01242 xx

026 xx

6x 2x

2,6 x

2

xxx 51212

The cutting board is a rectangle because of the reference to “long and wide.” Build a rectangle.

W

LWe know that the Length is twice the Width.

W2

The Area formula is L * W and the Area equals 800.

AreaWL

8002 WW

8002 2 W

08002 2 W

04002 2 W

020202 WW

20W 20W WL 240202 L

The dimensions are 40 cm by 20 cm.

Means that the numbers differ by 1. The First number is unknown, call it x.The Second number must be 1 bigger… x + 1

1561 xx

Multiply the two numbers and set = 156

15612 xx

015612 xx

01213 xx

13x 12x

Assuming the racing number must be positive, the first number is 12 and the consecutive second number is 13.

4402 xx44022 xx

044022 xx

Means that the numbers differ by 2. The First number is unknown call it x.The Second number must be 2 bigger… x + 2

02022 xx

22x 20x

There are two sets of answers!

-22 and -20

20 and 22

Right triangles have a special relationship called The Pythagorean Theorem. 222 cba a

b

c

The legs of the right triangle are the sides of the right angle, labeled a and b. The hypotenuse is the longest side and is labeled c.

x

3x

15222 cba

222 153 xx

22593322 xxxx

021662 2 xx

010832 2 xx

09122 xx

12x 9x

9

1239

The other two sides are 9 ft and 12 ft.

210N

21010010 2 tt

021010010 2 tt

0211010 2 tt

07310 tt

3t 7t

There will be 210 micrograms in the bloodstream at 3 minutes and 7 minutes.

135

100120

35

222 12035 c2144001225 c

215625 c

156250 2 c 1251250 cc

125c

c

The minimum length of the cable is 125 ft.

50h

10h

222 5010 hh

2500100101022 hhhh

02400202 2 hh

01200102 2 hh

030402 hh

30h

The two distances are 30 ft. and 40 ft.

A number is 6 less than its square. Find all such numbers.

x 62 x

60 2 xx

320 xx

2x 3x

62 xx

642

622 2

62 xx

693

633 2