## Inopeki 3 years ago TuringTest, want to continue?

1. TuringTest

lol I like the attitude, I better eat though, give me a bit what time is it where you are?

2. Inopeki

9:45 pm, you? Then go eat :)

3. TuringTest

alright, just so you know it's easy to send messages via fan message now, so use that instead of a question to hail me

4. Inopeki

Oh, ill use that!

5. TuringTest

see you in a bit

6. Inopeki

Alright

7. No-data

How do you use those fan messages Turing?

8. Inopeki

Go to the profile and click "write a fan message"

9. TuringTest

factor 21x^2+14x

10. Inopeki

GCF of 21 and 14=7 GCF of x and x^2 is x 21x^2/7x+14x/7x=x(3x+2)?

11. TuringTest

almost but 1)be careful of notation 21x^2/7x+14x/7x is not equal x(3x+2) 2) remember to keep the whole GCF outside the parentheses

12. Inopeki

But i thought that if the 7 was on the LHS then it couldnt be on the RHS?

13. Inopeki

21x^2/7x+14x/7x=7x(3x+2)?

14. TuringTest

but they are not equal 21x^2/7x+14x/7x=3x+2 not 7x(3x+2) so you must write 21x^2+14x=7x(21x^2/7x+14x/7x)=?

15. Inopeki

I dont know...

16. Inopeki

7(3x^2+2)?

17. TuringTest

you almost have it but you need to put the GCF outside the parentheses what is the GCF?

18. Inopeki

7x

19. TuringTest

so that's what goes outside the parentheses

20. Inopeki

7x(3x^2+2) Cant be right.

21. TuringTest

what is 21x^2/7x=? rewrite it in a way more comfortable to you if it looks strange

22. Inopeki

21/7 * x^2/x?

23. TuringTest

yes...

24. Inopeki

Im pretty sure thats 3x man

25. TuringTest

it is, so why did you write 3x^2 above?

26. Inopeki

Cause you said the other one wasnt right :/ What is the correct way to write it?

27. TuringTest

here was your first answer x(3x+2) here was the next 21x^2/7x+14x/7x=7x(3x+2) the answer here is correct, I just pointed out that the two expressions above are not equal, I think that may have confused you. next you wrote 7x(3x^2+2) so I think we started to get off track... the answer is 21x^2+14x=7x(3x+2) I just made a point about how you show your work, and what it means.

28. Inopeki

Ohhhhh :)

29. TuringTest

write out the process as 21x^2+14x=7x(3x)+7x(2)=7x(3x+2) that is really the best way to show factoring

30. Inopeki

Oh ok :D Another one?

31. TuringTest

factor 3t^3+9t^2

32. TuringTest

sorry, so you know how to divide t^3/t^2 ???

33. Inopeki

x*x*x ----- right? It should jjust become x? x*x

34. Inopeki

t, sorry

35. TuringTest

you got it, so my question stands: factor 3t^3+9t^2

36. Inopeki

GCF of 3 and 9 is 3. GCF of t^2 and t^3 is t. 3t^3+9t^2=3t(3t^3/3t)+3t(9t^2/3t)=3t(t^3+3t^2)?

37. TuringTest

check your answer and distribute to see if you get the original: $3t(t^3+3t^2)=3t(t^3)+3t(3t^2)=3t^4+9t^3\neq3t^3+9t^2$you didn't divide the terms in the middle right ...but I think more importantly the GCF of t^3 and t^2 is t^2, not t that is because each term can be evenly divided by t^2.

38. Inopeki

Hm

39. Inopeki

Oh right!

40. TuringTest

what is the GCF of t^3+t^5+t^7 ??

41. Inopeki

Something to the power of something doesnt follow the rules of usual numbers, forgot that. It should be t^5

42. TuringTest

no it's t^3 and it does follow the rules of regular numbers if you think about it, otherwise it wouldn't be true! look at 16,32,8 what is their GCF ?

43. Inopeki

8

44. TuringTest

now rewrite those three numbers as powers of 2

45. TuringTest

can you do that?

46. Inopeki

256,1024,64

47. Inopeki

?

48. TuringTest

no I meant like 8=2^3 16=2^? 32=2^?

49. Inopeki

Oh! 8=2^3 16=2^4 32=2^5

50. TuringTest

...and what is their GCF? 2^3 so the GCF of a set of variables is the highest common power to each term -in this case 3 so if we have y^4+y^7+y^9 the GCF is...?

51. Inopeki

y^4?

52. TuringTest

right :)

53. Inopeki

:D

54. TuringTest

good job :D so now back to our question... factor 3t^3+9t^2 what is the GCF?

55. Inopeki

3t^2?

56. TuringTest

right now can you factor it?

57. Inopeki

3t^3+9t^2=3t^2(3t^3/3t^2)+3t^2(9t^2/3t^2)=3t^2(t+3)?

58. TuringTest

very nice!!!!!!

59. Inopeki

I got it!! :D

60. TuringTest

you totally did :D that is very cool! so now lets see what's so great about factoring say we have f(t)=3t^3+9t^2 and we want to know the 'zeros' of the function that means when the function touches the x-axis, i.e. when f(t)=0 so to answer this we must solve 0=3t^3+9t^2 how can we solve that? by factoring... 0=3t^2(t+3) now you can solve it quickly, any idea which fundamental rule of algebra tells us how?

61. Inopeki

Nope, sorry

62. TuringTest

the rule is called the 'zero factor property' it states that if$ab=0$then either$a=0$or$b=0$or both does this rule make logical sense to you?

63. Inopeki

Yeah

64. TuringTest

so now look at our factored equation $0=ab=3t^2(t+3)$that means that either$3t^2=0$or $t+3=0$can you solve each of these equations?

65. TuringTest

*either or both I should say

66. Inopeki

OH so 3t^2 represents a and t+3 represents b!

67. TuringTest

exactly that's why i said to learn the rules on this page http://www.capitan.k12.nm.us/teachers/shearerk/basic_rules_of_algebra.htm almost all of algebra is in there, though sometimes it is hidden

68. Inopeki

3t^2=0 t=0-3t/t t=0-3 t=-3?

69. TuringTest

no, it's more simple than that, remember the same rule we just used: ab=0 then either a=0 or b=0 or both 3t^2=0 let 3=a t^2=b we know that a=3 cannot be zero, because 3 is never zero, it's a constant that leaves the possibility only of t^2=0 and the number number that times itself is zero is zero so$3t^2=0\to t=0$

70. Inopeki

Ohhhh.. Cause 3x0x0=0?

71. TuringTest

right so t has to be zero...

72. Inopeki

Yeah

73. TuringTest

what about the other possibility t+3=0 ???

74. Inopeki

How can that be possible when t is 0?

75. TuringTest

there are two answers to every quadratic equation as you may recall me saying this is cubic so it has 3 actually, we say that zero ocurrs twice in 3t^2=0 because it leads to two 0's as you showed: 3x0x0 we call that a 'multiplicity of 2' so there will be multiple answers, the other is found by solving t+3=0 remember that either a=0 or b=0 or BOTH we don't know so we have to solve them all.

76. Inopeki

Oh right! This is a quadratic equation. t=-3 on this one so its (0,-3)?

77. TuringTest

78. Inopeki

79. Inopeki

So these are cubic coordinates?

80. Inopeki

(0,0,-3)?

81. TuringTest

not cubic coordinates (I don't know what that is exactly...), it's just that we say that 3t^3+9t^2 has zeros (0,3) - that means the graph of f(x)=3t^3+9t^2 hits zero there... where 0 here has a 'multiplicity' of 2 (that means it occurs twice) and 3 has a multiplicity of 1

82. Inopeki

So why isnt -3 involved?

83. TuringTest

sorry typo, meant (0,-3)* and -3 has multiplicity 1*

84. TuringTest

good catch

85. Inopeki

Thanks

86. TuringTest

so do you see what I mean? the zero's of$f(t)=3t^2+9t^2$are found by factoring and setting to zero$0=3t^2(t+3)$then solving each possibility$3t^2=0\to t=0$$t+3=0\to t=-3$where we say that for t=0 k=2 and for t=-3 k=1 where k is the multiplicity

87. Inopeki

Right, the k is cause t is to the power of 3 there.

88. TuringTest

for the part that had 3t^2=0 we had k=2 (because zero is the answer twice: 3x0x0) for t+3=0 we have k=1 because t is only to the first power and we only have one answer If what you mean is that you noticed that adding up all the k's gave youu 3, the order of the cubic, then you have noticed what is called the Fundamental Theorem of Algebra: "The sum of the multiplicities of the roots of a function is equal to the order of the polynomial" $k_0+k_{−3}=2+1=3$

89. TuringTest

- a very important theorem as the name implies...

90. Inopeki

I dont really get the theorem, please explain it.

91. TuringTest

Basically for whatever the highest power variable you have in a polynomial, that is how many answers you have. Note that they may not be all different answers, but the ones that occur more than once are counted as having a higher multiplicity, so if you add up the multiplicities (the k's) that's how many zeros the polynomial has. for example 7x^5+3x^3+2x^2+5x+3=0 must have 5 answers, because it is 5th order x^2+2x+2=0 must have 2 answers, because it is second order

92. Inopeki

So x^4+5x-4=0 must have 4 answers?

93. TuringTest

exactly, though it may not be 4 different answers for instance x^4=0 has only the answer x=0, but that zero has a multiplicity of k=4 because 0x0x0x0=0 is how it must be...

94. Inopeki

Oh, now i get it

95. TuringTest

good :) do you want to try some more factoring problems? or perhaps you should learn a little about exponents first? or perhaps you are ready top get some rest.... which is it?

96. Inopeki

I wanna learn something new :D

97. TuringTest

let's see if this is news to you: simplify${x^{14}\over x^{12}}$

98. Inopeki

x^2 lol

99. TuringTest

good that saves a lot of time...

100. TuringTest

simplify$\sqrt[3]{x^{21}}$

101. Inopeki

x^7?

102. TuringTest

good :) more time saved...

103. Inopeki

:D actually i didnt know that

104. TuringTest

the genereal rule is$\sqrt[b]{x^{a}}=x^{a/b}$any radical can be wriitten as a fractional exponent, for instance$\sqrt x=x^{1/2}$so... let's try some of that.

105. Inopeki

Yeah, i assumed that :)

106. TuringTest

good guess, try to really rationalize it if you can... since you seem to know the rules let's try a trickier one simplify $\frac{\sqrt[3]{x^2}\sqrt[5]{x^3}}{\sqrt x}$

107. Inopeki

x^2/3 * x^3/5 ----------- Its basically this, right? If so, i just need to make the variables "suitable" to x^1/2 be merged, right?

108. TuringTest

right, just remember that$x^ax^b=x^{a+b}$and that$\frac{x^a}{x^b}=x^{a-b}$so you're gonna have and subtract to add those fractions, so they all need a common denominator.

109. TuringTest

so you're gonna have to add and subtract those fractions*

110. Inopeki

Yeah, thats what i meant by making them suitable to merge

111. TuringTest

I figured, but that's not a known term to me, just making sure

112. Inopeki

x^2/3 * x^3/5 x^10/15 * x^9/15 x^19/15 ----------- = =-------------- x^1/2 x^1/2 Now i need to get 19/15 divisible by 2

113. TuringTest

good so far :)

114. Inopeki

19/15 ------------38/30

115. Inopeki

x^38/30 -------- = x^1/2 Actually, dont i need to get them divisible by 15 so i can get 2 as the denominator?

116. TuringTest

what can you multiply the bottom by to get 1/2 over 30 ?

117. Inopeki

Oh! x^38/30 -------- = 23/30? No, that cant be it.. x^15/30

118. TuringTest

yes it can and is but you have to leave the x of course... x^(23/30)

119. Inopeki

Oh right lol

120. TuringTest

so that was good, I won't test you on that do you know how to FOIL ?

121. Inopeki

What?

122. TuringTest

i.e. simplify (a+b)(c+d)

123. Inopeki

abcd?

124. TuringTest

FOIL= First Outer Inner Last|dw:1325979609902:dw|watch the arrows they multiply the first in the brackets, the outer terms, the inner, and the last seperateyl

125. TuringTest

it's just like distribution, but you have to do a and b seperate

126. Inopeki

I think i get it. give me another one and ill solve it fast

127. TuringTest

It's okay if you don't solve it fast actually remember that solving is not simplifying, here we have no = sign so we are simplifying: (x+3)(x+2)

128. Inopeki

x^2+2x+3x+6?

129. TuringTest

nice! you can simplify the middle terms

130. Inopeki

Oh right,x^2+5x+6

131. TuringTest

nicely done! especially for never having FOILed before. simplify (2x+5)(3x-1)

132. Inopeki

5x+2x+15x-5?

133. Inopeki

6x*

134. TuringTest

careful... what's the first? what's the outer? the others are right.

135. TuringTest

yes the first is 6x the outer though...

136. Inopeki

-1?

137. TuringTest

times what?

138. Inopeki

2x, maybe thats -2x..

139. TuringTest

there ya go ;-)

140. TuringTest

so it should be...?

141. Inopeki

6x+-2x+15x-5

142. TuringTest

scratch the extra plus sign, but yes, now simplify...

143. Inopeki

29x-5?

144. TuringTest

oh my mistake, you forgot that the first term would be squared First=2x(3x)=6x^2 now what do you get?

145. Inopeki

6x^2-2x+15x-5?

146. TuringTest

yes, and now simplify

147. Inopeki

4x^2+15x-5?

148. TuringTest

like terms, x with x.... not x with x^2...

149. Inopeki

so 6x^2+13x-5?

150. TuringTest

yes. good job. now simplify (2p-3)(5p-1)

151. Inopeki

10p^2-2p-15p-3? Simplified:10p^2-17p-3

152. TuringTest

good, but watch the last term negative times negative is...?

153. Inopeki

Oh right, positive. So 10p^2-17p+3?

154. Inopeki

2r^2+2r*3t+t*r+3t^2 Simplified: 2r^2+2r^2+3t^2+3t^2

155. TuringTest

stop before your simplification (which is wrong, sorry) look at the first step: 2r^2+2r*3t+t*r+3t^2 ^^^ should everything be positive?

156. Inopeki

Oh right.. 2r^2+2r*-3t+t*r

157. TuringTest

yes but you forgot the last term you should also put parentheses around the negative terms

158. Inopeki

2r^2+2r*(-3t)+t*(-r)?

159. TuringTest

lol it got deleted, what was the original problem please?

160. Inopeki

I cant remember :(

161. TuringTest

ok another, because we started losing terms in the last one (2y+x)(4y-3x)

162. Inopeki

8y+2y*3x+x*4y(-3x^2)

163. TuringTest

there should be four terms what is your first? write out the middle step

164. TuringTest

it's close in many ways, but there are quite a few mistakes

165. Inopeki

Ill put them in prenthesis (8y)+(2y*3x)+(x*4y)-(3x^2)

166. TuringTest

better, but the first is 2y(4y) no? and the outer is a positive times a negative as well

167. Inopeki

Oh, ill solve this one and then sleep :) (8y^2)+(2y*-3x)+(x*4y)-(3x^2)

168. TuringTest

nice, so to simplify this first multiply the coefficients...

169. Inopeki

(8y^2)+(8y^2*-3x^2)-(3x^2)

170. TuringTest

not I think you're getting tired (8y^2)+(2y*-3x)+(x*4y)-(3x^2)=8y^2-6xy+4xy-3x^2=?

171. TuringTest

no*

172. TuringTest

can you simplify the last step?

173. Inopeki

Yeah, i need to sleep.

174. TuringTest

for future reference:$(8y^2)+(2y*-3x)+(x*4y)-(3x^2)$$=8y^2-6xy+4xy-3x^2=8y^2-2xy-3y^2$goodnight!

175. Inopeki

Good night! Thanks for all youve taught me :D See you tomorrow?