Oh I finished!
You can check those links to understand the concept of Probability
http://en.wikipedia.org/wiki/Probability
http://en.wikipedia.org/wiki/Probability_theory
See the Strategy for Finding the GCF for Monomials- show work- the steps
40. 16x2z, 40xz2, 72z3
16 = 24
40 = 235
72 = 2332
GCF of 16, 40 and 72 is 23 =8
GCF of x2z, xz2 and z3 is z because the common variable is z and the smallest power of z=1
Answer: 8z
Factor out the GCF in each expression
68. 15x2y2 - 9xy2 + 6x2y
GCF of 15x2y2 , 9xy2 , 6x2y is 3xy
Then 15x2y2 - 9xy2 + 6x2y=3xy(5xy-3y+2x)
Answer: 3xy(5xy-3y+2x)
Factor out the GCF in each expression
72. a(a+ 1) -3(a+ 1)
a(a+ 1) -3(a+ 1)=(a+1)(a-3)
Answer: (a+1)(a+3)
Section 5.2
Factor each polynomial.
16. 9a2- 64b2
9a2-64b2 = (3a)2-(8b)2 = (3a+8b)(3a-8b)
Answer: (3a+8b)(3a-8b)
Factor each polynomial completely See example 4.
62. x3y + 2x2y2 + xy3
The GCF of x3y, 2x2y2 , xy3 is xy then
x3y + 2x2y2 + xy3 = xy(x2+2xy+y2) and (x2+2xy+y2) =(x+y)2
then: x3y + 2x2y2 + xy3 = xy(x+y)2
Answer: xy(x+y)2
Use grouping to factor each polynomial completely; see example 5
80. x3 + ax + 3a + 3x2
3
=x
+ ax + 3a + 3x2 = (x3 + ax) +( 3a + 3x2)= (x3 + ax) +( 3x2+3a)= x(x2+a)+3(x2+a)= (x+3)(x2+a)
2
Answer: (x+3)(x +a)
Section 5.3
Factor each trinomial. If the polynomial is prime, say so. See examples 2-4
58. 18z + 45 + z2
We need to factor z2+18z+45, we need to find 2 integers with a product of 45 and a sum of 18
If the product is positive and the sum is positive then those numbers are positive
Those numbers are 3 and 15 then
z2+18z+45 = z2+3z+15z+45 = (z2+3z)+(15z+45) = z(z+3)+15(z+3)=(z+15)(z+3)
Answer: (z+15)(z+3)
Factor each polynomial. See example 5
64. h2 - 9hs + 9s2
To factor h2 - 9hs + 9s2 we need to find 2 integers with a product of 9 and a sum of -9
If the product is 9 then there are 2 positive numbers or 2 negative numbers, since the sum is -9
We must find find 2 negative integers with product 9 and sum -9
If the product is 9 then numbers are -3 and -3 or -1 and -9 but the sum is -6 or -10
Then we can´t factor this polynomial
Answer: is prime
Factor each polynomial completely. Use the methods discussed
in Sections 5.1 through 5.3. If the polynomial is prime say so.
102. 3x3y2 - 3x2y2 + 3xy2
The GCF of 3x3y2 , 3x2y2 ,3xy2 is 3xy2
Then: 3x3y2 - 3x2y2 + 3xy2=3xy2(x2-x+1), x2-x+1 is prime
Answer: 3xy2(x2-x+1)
Section 5.4
Factor each trinomial using the ac method.
See Example 1.
See the Strategy for Factoring ax2 + bx +c by the ac Method box on page 344
18. 2x2 +11x + 5
a=2, b=11, c=5
We must find 2 numbers that have a product of ac=10 and a sum of b=11
Those numbers are positive since the product and the sum are positive
These numbers are 1 and 10
2x2 +11x + 5 = 2x2 +(1+10)x + 5 = 2x2 +x+10x + 5 = x(2x+1)+5(2x+1)=
(x+5)(2x+1)
Answer: (x+5)(2x+1)
26. 21x2 + 2x – 3
a=21, b=2, c=-3
We must find 2 numbers that have a product of ac=-63 and a sum of b=2
These numbers are -7 and 9
21x2 +2x -3 = 21x2 +(-7+9)x -3 = 21x2 -7x+9x-3 = 7x(3x-1)+3(3x-1)=
(7x+3)(3x-1)
Answer: (7x+3)(3x-1)
38. 8x2 - 10x – 3
a=8, b=-10, c=-3
We must find 2 numbers that have a product of ac=-24 and a sum of b=-10
These numbers are 2 and -12
8x2 -10x -3 = 8x2 +(2-12)x -3 = 8x2 +2x-12x-3 =(8x2+2x)-(12x+3)= 2x(4x+1)-3(4x+1)=
(2x-3)(4x+1)
Answer: (2x-3)(4x+1)
Factor each polynomial completely. See examples 5 and 6
88. a2b +2ab -15b
GCF of a2b ,2ab , -15b is b
Then a2b +2ab -15b = b(a2+2a-15)
(1)
To factor a2+2a-15 we must find 2 integers with a product of -15 and a sum of 2
Those integers are 5 and -3
a2+2a-15 = a2+(5-3)a-15= a2+5a-3a-15 = a(a+5)-3(a+5)=(a-3)(a+5) (2)
From (1) and (2)
Answer: b(a-3)(a+5)
Section 5.5
Factor each polynomial completely
36. m4 - n4
m4 - n4 = (m2)2 – (n2)2 = (m2+n2)(m2-n2)= (m2+n2)(m+n)(m-n)
Answer: (m2+n2)(m+n)(m-n)
Factor each polynomial completely. If a polynomial is prime,
say so.
See example 4
See the Strategy for factoring Polynomials completely box on page 355
44. 3x3 - 12x
GCF of 3x3 , 12x is 3x then 3x3 - 12x=3x(x2-4)=3x(x2-22)=3x(x+2)(x-2)
Answer: 3x(x+2)(x-2)
66. 8b2 + 24b + 18
GCF of 8b2 , 24b , 18 is 2 then
8b2 + 24b + 18 = 2(4b2+12b+9)=2((2b)2+2(2b)3+32)=2(2b+3)2
Answer: 2(2b+3)2
72. 3x2 -18x – 48
Gcf of 3x2 ,18x , 48 is 3 then 3x2 -18x – 48 = 3(x2-6x-16)
(1)
To factor x2-6x-16 we must find 2 integers where the product is -16 and the sum is -6
Those numbers are -8 and 2
x2-6x-16 = x2+(2-8)x-16=x2+2x-8x-16=(x2+2x)-(8x+16)=x(x+2)-8(x+2)=(x-8)(x+2) (2)
From (1) and (2)
Answer: 3(x-8)(x+2)
82. 9x2 + 4y2
9x2 + 4y2 = (3x)2+(2y)2 is prime ( a2+b2 is prime)
Answer: is prime
Section 5.6
Solve by factoring.
18. 2h2 – h- 3 = 0
To factor 2h2 – h- 3 (a=2,b=-1,c=-3) using the ac method
We have to find 2 numbers where the product is ac=-6 and the sum is b=-1
Those numbers are -3 and 2
2h2 +(2-3)h- 3 = 2h2+2h-3h-3 = 2h(h+1)-3(h+1)=(2h-3)(h+1)
We must solve (2h-3)(h+1)=0 then 2h-3 =0 or h+1=0
Then h=3/2 or h=-1
Answer: -1,3/2
Solve each equation. See examples 2 and 3
32. 2w(4w + 1) = 1
8w2+2w=1
8w2+2w-1=0
To factor 8w2+2w-1(a=8,b=2,c=-1) using the ac method
We have to find 2 numbers where the product is ac = -8 and the sum is b=2
Those numbers are 4 and -2
8w2 +(4-2)w- 1 = 8w2+4w-2w-1 = 4w(2w+1)-1(2w+1)=(4w-1)(2w+1)
We must solve (4w-1)(2w+1)=0 then 4w-1 =0 or 2w+1=0
Then w=1/4 or w=-1/2
Answer: ¼, -1/2
Solve each equation.
54. x2 - 36 = 0
x2 – 36 = x2-62= (x+6)(x-6)=0
then x+6=0 or x-6=0 then x=-6 or x=6
Answer: -6,6
58. x3 = 4x
x3=4x then x3-4x=0 then x(x2-4)=0 then x(x+2)(x-2)=0
then x=0 or x+2=0 or x-2=0 then
x=0 or x=-2 or x=2
Answer: 0,-2,2
66. (x - 3)2
(x - 3)2
+
+
(x + 2)2 = 17
(x + 2)2 = x2-6x+9+x2+4x+4=2x2-2x+13
So we must solve 2x2-2x+13=17
2x2-2x-4=0 then
2(x2-x-2)=0 then
2(x-2)(x+1)=0 then
x-2 =0 or x+1=0 then
x=2 or x=-1
Answer: 2,-1
98. Avoiding a collision. A car is traveling on a road that
is perpendicular to a railroad track. When the car is
30 meters from the crossing, the car’s new collision
detector warns the driver that there is a train 50 meters
from the car and heading toward the same crossing. How
far is the train from the crossing?
Let x = distance between the train and the crossing
The triangle formed by the car position,train position and the crossing it,s a right triangle
Then 502=302+x2
x2= 502-302 =2500-900=1600
x =1600 = 40
Answer: 40 meters
100. Winter wheat. While finding the amount of seed needed
to plant his three square wheat fields, Hank observed that
the side of one field was 1 kilometer longer than the side
of the smallest field and that the side of the largest field
was 3 kilometers longer than the side of the smallest field.
If the total area of the three fields is 38 square kilometers,
then what is the area of each field?
Let x= the side of the smallest field
Sides of the fields are x,x+1 and x+3
Total area is x2+ (x+1)2+(x+3)2 = 3x2+8x+10
Then we must solve 3x2+8x+10 = 38
3x2+8x-28 = 0
We must find 2 numbers where the product is ac = -84 and the sum is b=8
Those numbers are 14 and -6
3x2+(14-6)x-28=0
3x2+14x-6x-28=0
(3x2+14x)-(6x+28)=0
x(3x+14)-2(3x+14)=0
(x-2)(3x+14)=0
x-2 =0 or 3x+14=0
x=2 or x=-14/3
But x must be positive (a negative side has no sense)
Then the smallest side is 2 meters
Answer: 2,3 and 5 meters
104. Venture capital. Henry invested $12,000 in a new
restaurant. When the restaurant was sold two years
later, he received $27,000. Find his average annual
return by solving the equation
12,000(1 + r)2 =27,000.
12,000(1 + r)2 =27,000
(1+r)2 = 27,000/12,000= 27/12=9/4
(1+r) = ±(9/4) = ±3/2
r = 3/2-1 or -3/2-1 but r must be positive
Then r = 3/2-1=0.5
Answer: r = 0.5 (50%)
Notes:
Tell me if you have any doubt about my answers
It was a long work, I hope receiving a good bonus as the last time
Thanks
Steve