Download Factoring Package Day 2

factoring whole numbers – factor – write as a product of 2 or more numbers
factor fully – factor so that none of the factors can themselves be further factored
1, 2, 3, 5, 7, 11, 13, 17, 19, 23… etc. can’t be further factored (i.e. prime numbers)
1) 4 = 2 x 2
7)
12 = 2 x 6 = 2 x 2 x 3
2) 10 = 2 x 5
8) 18 = 2 x 9 = 2 x 3 x 3
3) 15 = _________
9) 8 = _____________________________________________
4) 9 = __
10) 48 = 2 x 24 = 2 x 2 x 12 = 2 x 2 x 2 x 6 = 2 x 2 x 2 x 2 x 3
5) 6 = __
11) 24 = __ = ________________________
6) 21 = _
12) 30 = _ = __________________________________
factoring monomials
1) a2 = a x a
6) 2a2 = 2 x a x a
11) 9a2 = 3 x 3 x a x a
16) 8a2 = 2 x 2 x 2 x a x a
2) b3 = b x b x b
7) 11b2 = 11 x b x b
12) 15b3 = 3 x 5 x b x b x b
17) 12b2 = 2 x 2 x 3 x b x b
3) c2 =
8) 3c3 =
13)
18) 18c3 =
4) d3 =
9) 5p2 =
14) 6d
19) 20d2 =
5) k2 =
10) 7r3 =
15)
20) 24e2 =
common factoring
steps: A)
B)
C)
D)
E)
factor each term
find the factors that are common for all the terms
multiply these common factors to find the greatest common factor
use the distributive law to rewrite
use the distributive law to check your answer
e.g 1) factor: 4x3 + 6x2
e.g. 2) factor: 6x2 – 9x + 15
e.g. 3) factor: 12x2 + 15x
A)
4x3 = 2 · 2 · x · x · x
6x2 = 2 · 3 · x · x
6x3 = 2 · 3 · x · x
9x = 3 · 3 · x
15 = 3 · 5
12x2 = 2 · 2 · 3 · x · x
15x = 3 · 5 · x
B)
2, x, and x are common to both terms
3 is the only common factor
3, and x are common to both terms
C)
2 · x · x = 2x2
so, greatest common factor is 2x2
so, greatest common factor is 3
3 · x = 3x
so, greatest common factor is 3x
D)
2x2(2x + 3)
3(2x2 – 3x + 5)
3x(4x + 5)
E)
2x2(2x + 3) = 4x3 + 6x2
3(2x2 – 3x + 5) = 6x2 – 9x + 15
3x(4x + 5) = 12x2 + 15x
Factoring – Common Factoring binomials
COMMON factoring (common factoring binomials)
1)
y = 2x + 4
y = 2(x + 2)
2)
y = 3x + 9
y=
3)
y = 5x + 15
y=
4)
y = 4x + 20
y=
5)
y = 2x – 8
y=
6)
y = 3x – 12
y=
7)
y = 5x – 10
y=
8)
y = 4x – 16
y=
9)
y = –2x + 4
y=
10) y = –3x + 9
y=
11) y = –5x + 15
y=
12) y = –4x + 20
y=
13) y = –2x – 8
y=
14) y = –3x – 12
y=
15) y = –5x – 10
y=
16) y = –4x – 16
y=
17) y = 2x2 + 4x
y = 2x(x + 2)
18) y = 3x2 + 9x
y=
19) y = 5x2 + 15x
y=
20) y = 4x2 + 20x
y=
21) y = 2x2 – 8x
y=
22) y = 3x2 – 12x
y=
23) y = 5x2 – 10x
y=
24) y = 4x2 – 16x
y=
25) y = –2x2 + 4x
y=
26) y = –3x2 + 9x
y=
27) y = –5x2 + 15x
y=
28) y = –4x2 + 20x
y=
29) y = –2x2 – 8x
y=
30) y = –3x2 – 12x
y=
31) y = –5x2 – 10x
y=
32) y = –4x2 – 16x
y=
check your answer by expanding…
y = 2x + 4
check your answer by expanding…
2x2 + 4x
Factoring – Common Factoring trinomials
COMMON factoring
continued
page 2
(common factoring trinomials)
1)
y = 2x2 + 6x + 8
y = 2(x2 + 3x + 4)
2)
y = 3x2 + 9x + 15
y=
3)
y = 4x2 + 12x + 8
y=
4)
y = 5x2 + 15x + 10
y=
5)
y = 2x2 + 8x – 12
y=
6)
y = 3x2 + 12x – 9
y=
7)
y = 4x2 + 16x – 12
y=
8)
y = 5x2 + 25x – 15
y=
9)
y = 2x2 – 6x + 8
y=
10) y = 3x2 – 9x + 15
y=
11) y = 4x2 – 12x + 8
y=
12) y = 5x2 – 15x + 10
y=
13) y = 2x2 – 8x – 12
y=
14) y = 3x2 – 12x – 9
y=
15) y = 4x2 – 16x – 12
y=
16) y = 5x2 – 25x – 15
y=
17) y = –2x2 + 6x + 8
y = –2(x2 – 3x – 4)
18) y = –3x2 + 9x + 15
y=
19) y = –4x2 + 12x – 8
y=
20) y = –5x2 + 15x – 10
y=
21) y = –2x2 – 8x + 12
y=
22) y = –3x2 – 12x + 9
y=
23) y = –4x2 – 16x – 12
y=
24) y = –5x2 – 25x – 10
y=
check your answer by expanding… y = 2x2 + 6x + 8
check your answer by expanding… y = –2x2 + 6x + 8
COMMON factoring (common factoring binomials)
1)
y = 2x + 4
y = 2(x + 2)
2)
y = 3x + 9
y = 3(x+3)
3)
y = 5x + 15
y = 5(x+3)
4)
y = 4x + 20
y = 4(x+5)
5)
y = 2x – 8
y = 2(x+4)
6)
y = 3x – 12
y = 3(x-4)
7)
y = 5x – 10
y = 5(x-2)
8)
y = 4x – 16
y = 4(x-4)
9)
y = –2x + 4
y = -2(x-2)
10) y = –3x + 9
y = -3(x-3)
11) y = –5x + 15
y = -5(x-3)
12) y = –4x + 20
y = -4(x-5)
13) y = –2x – 8
y = -2(x+4)
14) y = –3x – 12
y = -3(x+4)
15) y = –5x – 10
y = -5(x+2)
16) y = –4x – 16
y = -4(x+16
17) y = 2x2 + 4x
y = 2x(x + 2)
18) y = 3x2 + 9x
y = 3x(x+3)
19) y = 5x2 + 15x
y = 5x(x+3)
20) y = 4x2 + 20x
y = 4x(x+5)
21) y = 2x2 – 8x
y = 2x(x-4)
22) y = 3x2 – 12x
y = 3x (x-4)
23) y = 5x2 – 10x
y = 5x(x-2)
24) y = 4x2 – 16x
y =4x(x-4)
25) y = –2x2 + 4x
y = -2x(x-2)
check your answer by expanding…
y = 2x + 4
check your answer by expanding…
2x2 + 4x
26) y = –3x2 + 9x
y = -3x(x-3)
27) y = –5x2 + 15x
y = -5x(x-3)
28) y = –4x2 + 20x
y = -4x(x-5)
29) y = –2x2 – 8x
y = -2x(x+4)
30) y = –3x2 – 12x
y = -3x(x+4)
31) y = –5x2 – 10x
y = -5x(x+2)
32) y = –4x2 – 16x
y = -4x(x+4)
Factoring – Common Factoring trinomials
COMMON factoring
page 2
continued (common factoring trinomials)
y = 2x2 + 6x + 8
y = 2(x2 + 3x + 4)
y = 3x2 + 9x + 15
y = 3(x2 + 3x+ 5)
y = 4x2 + 12x + 8
y=
check your answer by expanding… y = 2x2 + 6x + 8
4(x 2+ 3x+ 2)
y = 5x2 + 15x + 10 y =
5(x 2 + 3x+ 2)
y = 2x2 + 8x – 12
y=
2( x 2+ 4x− 6)
y = 3x2 + 12x – 9
2
y = 3(x + 4x− 3)
2
y = 4x2 + 16x – 12 y = 4(x + 4x− 3)
2
y = 5x2 + 25x – 15 y = 5(x + 5x− 3)
y = 2x2 – 6x + 8
2
y = 2( x − 3x+ 4)
y = 3x2 – 9x + 15
2
y = 3(x − 3x+ 5)
y = 4x2 – 12x + 8
2
y = 4 (x − 3x+ 2)
2
y = 5x2 – 15x + 10 y = 5(x − 3x+ 2)
y = 2x2 – 8x – 12
2
y = 2( x − 4x− 6)
y = 3x2 – 12x – 9
2
y = 3(x − 4x− 3)
2
y = 4x2 – 16x – 12 y = 4(x − 4x− 3)
2
y = 5x2 – 25x – 15 y = 5(x − 5x− 3)
y = –2x2 + 6x + 8
y = –2(x2 – 3x – 4)
y = –3x2 + 9x + 15 y =
− 3( x 2− 3x− 5)
2
y = –4x2 + 12x – 8 y = − 4( x − 3x+ 2)
y = –5x2 + 15x – 10 y =
− 5( x 2− 3x+ 2)
y = –2x2 – 8x + 12 y =
− 2(x 2 + 4x− 6)
y = –3x2 – 12x + 9 y =
− 3( x 2+ 4x− 3)
2
y = –4x2 – 16x – 12 y = − 4( x + 4x+ 3)
check your answer by expanding… y = –2x2 + 6x + 8
2
y = –5x2 – 25x – 10 y = − 5( x + 5x+ 2)