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2-7-1 Number Properties Properties are the rules of the game. If the rules are changed a different kind of math is played. These properties are for everyday math. Commutative property of addition. Commutative property of multiplication. a+b=b+a axb=bxa 7+8=8+7 (-3)+4=4+(-3) 7x8=8x7 (-3)4=4(-3) The commutative property says the order doesn’t matter for addition. The commutative property says the order doesn’t matter for multiplication. Associative property of addition. (a+b)+c = a+(b+c) (7+8)+2=7+(8+2) 15+2 = 7 + 10 Associative property of multiplication. (axb)xc=ax(bxc) (7x8)x2=8x(7x2) 56x2=8x14 The associative property says the grouping doesn’t matter for addition. The associative property says the grouping doesn’t matter for multiplication. An identity for a particular operation doesn’t change the identity of the number when the operation is done. 7 + 0 =7 Additive Identity a+0=a Zero plus a number is that number. 8x1=8 Multiplicative Identity 1xa=a One times a number is that number. An inverse for a particular operation is the number that returns the identity. Additive inverse a + (-a) =0 Multiplicative inverse 1 a 1 a The additive inverse is the negative of the number. 7 + (-7) =0 -7 is the additive inverse of 7 3 x 1/3 = 1 The multiplicative inverse of a rational number is it’s reciprocal. 3x(4x)=3(4)xx=12x2 using both the commutative and associative properties of multiplication. Distributive property a(b+c)=ab+ac 53 4 5 3 5 4 Multiplying a sum by some number is the same as multiplying each term by that same number. 3x2 x 8 6 x 2 24 x Practice: List the property illustrated by the following example. a) 5+0=5 R+6=6+R 3+(7+8)=(3+7)+8 b) 4t(7t8t)= (4t7t)8t c) xy=yx 4 4 4 9 2 4 (9) 4 2 0+t=t 2z+7=7+2z 3x=3x+0 1x=x 7(2x-8)=14x-56 d) m m 2 3 4 5 1 2 1 1 1 3 4 5 6 5 e) -8d+8d=0 0-7y=-7y 40 8+(-8)=0 9t 12 3 12t 16 4 15y+10=5(3y+2) 1 4 1 4 -2e(3e-7)=-6e2+14e 2-7-2 Distributive Property To illustrate the distributive property look at the problem 3(2x-7). This problem says 3 times 2x-7 or 3 of (2x-7). Multiplying by 3 means add 2x-7 over and over. 2x-7+2x-7+2x-7 Simplify this by using the commutative property. 2x+2x+2x-7-7-7=3(2x)+3(-7) To use the distributive property, multiply each term inside the parenthesis by the multiplier. Study the following examples. Look where exponent rules are used. Watch the multiplication when negatives are involved. After the distribution, combine any like terms. Examples: 5 3x 8 15x 40 3x 7 6 x 21x 18 x 2 5s 8 5s 8 4 x 2 3x 9 4 x 6 x 18 2 x 18 3w2 y3 7w 5 y 2 21w3 y3 15w2 y5 5 3 6x 5 3 6x 2 6x There can be more than two terms inside the parenthesis. 3 4 6d 8d 2 12 18d 24d 2 4 z 2 6 5 z 24 z 2 20 z 3 5 7 x2 5x 9 35x2 25x 45 6 xy 2 5x2 y 2 xy 3xy 2 30 x3 y3 12x2 y3 18x2 y 4 3gh3 7 gh 5gh3 6 gh2 3gh3 7 gh 5gh3 6gh2 8gh3 7 gh 6gh2 16 x 2 12 x 16 x 2 12 x 4 x 2 3x 4 4 4 16 x 2 12 x 16 x 2 12 x 4 x 2 3x 4 4 4 15w3 25w5 75w4 15w3 25w5 75w4 5w 2 1 5w 15w3 15w3 15w3 15w3 3 Practice: a) 3 2 x 5 25 2 y 3 7 x 8 b) 53 8x 3 3 7 x 100 2 5 x c) 4 2x 5x 8 4 2 y d) 4 3x 8 10 5 x 15 2 7 8y e) 5 y 7 6 y f) 3 j 6 j 5 j 3 8h 7h 6 2 g) 9 5a 8a 4m2 m 5 4l 12 2 2 3 k 12 5k 1 2 z 2 6 10 z u 8 41 h) 3x 3x 9 9h 2 5h 8 i) 2 y5 z 3 7 y 2z 2 j) 8 5 9x 5 2 3 3 7 8 9x k) 8 5 3x 9 9 5 x 5 13 7 3x 9h 9 3w 6 8w2 q 2 6q 8 l) 8 2 6 x 5 x m) 4 10 4y y 2 2 3 3 n) 2x y x y xy xy 2 3 o) uv u v uv uv 2 2 2 4 2 mn3 8m5 5n2 5 15x 2 4 3 16 x 2 12 x s3t 2 2 3 s 2t 2 6 5 st 2 3st 2 5 3 3 2 s) 2 2 a 2 2a 25 10 x 2 10 x 100 6 r) 4a b 10 gh 5g h gh 2 x 2 12 x 1 2 6e2 12e 2 30v 3 21v 5 18v8 3v 5 2 2 31 5 7u 2 8u 1 25 p 3 15 p 5 q) gh gh gh gh vw8 15v w v2 5 10 6 x 9 12 5x 9 1 p) 9x 7 x 5 7 12 3 xy3 3xy 3 4 xy3 xy 2 v2 w3 vw 38 v2 w3 38 vw2 15h3 25h51 75h 25 5h31 35k 5 21k 4 77k 7 7k 3 15u 3 5u 5 75u 4 3u 3i 5 j 3 21i 4 j 5 i 7 j 3 u) 3i 2 j 3 3x5 y 3 21x 4 y 5 12 x 7 y 3 9 x5 y 2 5w3v 3 w4v 4 w7v 3 v) wv 5 3e4 f 13 21e2 f 5 e5 f 5 e 2 f 5 t) 42 2-7-3 Factoring Assume 12x+6 is the result of distribution. What did it look like before the distribution was done? There was something outside a set of parenthesis and two terms inside the parenthesis. ( + ) 6 ( 2x + 1 ) Separating an expression so that the smallest possible pieces called factors multiply together to get the original expression is called factoring. Look at each term and find the largest factor that is in all terms. Example: 72 x 60 x 36 Find the largest number that divides evenly into 72, 60 and 36. This number goes in front of the parenthesis. 72 x 2 60 x 36 2 Divide each term by the common factor. Fill in the positions in the parenthesis. Distribute to check the factoring. 24x2 y 2 16x2 y 56xy 2 12 , , 12 12 ( 12 ( 12 6x2 + + ) + 5x + 3 ) The largest factor in all terms is 8xy. When the division is done there shouldn’t be any negative exponents. Notice the negative in the third term. Divide each term by the common factor. 8xy ( 24 x y 16 x y 56 xy , , 8 xy 8 xy 8 xy 2 2 2 + - ) + 2x - 7y ) 2 8xy ( 3xy The division can be done in your head, but some students need to write it down. Practice: Factor the following. a) 15y-25=5(3y-5) 14z+56= 39u-13= 81p-18= 7t+21= b) 8+8t= 14-21w= 9v-81= 2x-10= 3w+9= c) -2/3y-10/3=-2/3(y+5) Note the negative. 4y 8 = 5 5 10 15r = 12 12 d) 6x2-9x+12= 3(2x2-3x+4) 14e2+21e+56= 2x2-4x-8= 8a2+4a+4= 10b2+5b-25 -5-5t-5t2= -8+16e-24e2= 12-24y2+36y= 15r+15r2-25 bx-7x= 3s-3ts= 4ef+3f= 2w-wx= 14y2+21y= 81z+18z2= 10q2-20q= 8a-8a2= 12xy2+16xy+24x= a2x2+a2x+a2= 2a2x-4a= e3-5e2x= e) 3+3w+3w2= f) ax+2a=a(x+2) g) 4x2-12x=4x(x-3) h) 2ax2+4ax-8a= i) 3x3y2+9x2y2-12x2y3=3x2y2(x+3-4y) 4 25 y6 25 5r 2 3 3 25a4b2+45a2b3-15a2b5= 43 j) a7b5 5a8b7 a5b4 k) 25a4b2 + 75a2b3 - 100a2b2 l) 3 44 x 1 4 4x 4 2 Hint: Factor out a ¼. 2e4 f 3 4e2 f 2 ef 3 21x3y6+14x2y5-28x4y4 15a 2 21a 8 8