## Inopeki Group Title TuringTest, want to continue? 2 years ago 2 years ago

1. TuringTest Group Title

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

2. Inopeki Group Title

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

3. TuringTest Group Title

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 Group Title

Oh, ill use that!

5. TuringTest Group Title

see you in a bit

6. Inopeki Group Title

Alright

7. No-data Group Title

How do you use those fan messages Turing?

8. Inopeki Group Title

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

9. TuringTest Group Title

factor 21x^2+14x

10. Inopeki Group Title

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

11. TuringTest Group Title

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 Group Title

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

13. Inopeki Group Title

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

14. TuringTest Group Title

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 Group Title

I dont know...

16. Inopeki Group Title

7(3x^2+2)?

17. TuringTest Group Title

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

18. Inopeki Group Title

7x

19. TuringTest Group Title

so that's what goes outside the parentheses

20. Inopeki Group Title

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

21. TuringTest Group Title

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

22. Inopeki Group Title

21/7 * x^2/x?

23. TuringTest Group Title

yes...

24. Inopeki Group Title

Im pretty sure thats 3x man

25. TuringTest Group Title

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

26. Inopeki Group Title

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

27. TuringTest Group Title

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 Group Title

Ohhhhh :)

29. TuringTest Group Title

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 Group Title

Oh ok :D Another one?

31. TuringTest Group Title

factor 3t^3+9t^2

32. TuringTest Group Title

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

33. Inopeki Group Title

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

34. Inopeki Group Title

t, sorry

35. TuringTest Group Title

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

36. Inopeki Group Title

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 Group Title

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 Group Title

Hm

39. Inopeki Group Title

Oh right!

40. TuringTest Group Title

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

41. Inopeki Group Title

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

42. TuringTest Group Title

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 Group Title

8

44. TuringTest Group Title

now rewrite those three numbers as powers of 2

45. TuringTest Group Title

can you do that?

46. Inopeki Group Title

256,1024,64

47. Inopeki Group Title

?

48. TuringTest Group Title

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

49. Inopeki Group Title

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

50. TuringTest Group Title

...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 Group Title

y^4?

52. TuringTest Group Title

right :)

53. Inopeki Group Title

:D

54. TuringTest Group Title

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

55. Inopeki Group Title

3t^2?

56. TuringTest Group Title

right now can you factor it?

57. Inopeki Group Title

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

58. TuringTest Group Title

very nice!!!!!!

59. Inopeki Group Title

I got it!! :D

60. TuringTest Group Title

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 Group Title

Nope, sorry

62. TuringTest Group Title

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 Group Title

Yeah

64. TuringTest Group Title

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 Group Title

*either or both I should say

66. Inopeki Group Title

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

67. TuringTest Group Title

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 Group Title

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

69. TuringTest Group Title

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 Group Title

Ohhhh.. Cause 3x0x0=0?

71. TuringTest Group Title

right so t has to be zero...

72. Inopeki Group Title

Yeah

73. TuringTest Group Title

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

74. Inopeki Group Title

How can that be possible when t is 0?

75. TuringTest Group Title

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 Group Title

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

77. TuringTest Group Title

78. Inopeki Group Title

79. Inopeki Group Title

So these are cubic coordinates?

80. Inopeki Group Title

(0,0,-3)?

81. TuringTest Group Title

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 Group Title

So why isnt -3 involved?

83. TuringTest Group Title

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

84. TuringTest Group Title

good catch

85. Inopeki Group Title

Thanks

86. TuringTest Group Title

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 Group Title

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

88. TuringTest Group Title

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 Group Title

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

90. Inopeki Group Title

I dont really get the theorem, please explain it.

91. TuringTest Group Title

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 Group Title

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

93. TuringTest Group Title

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 Group Title

Oh, now i get it

95. TuringTest Group Title

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 Group Title

I wanna learn something new :D

97. TuringTest Group Title

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

98. Inopeki Group Title

x^2 lol

99. TuringTest Group Title

good that saves a lot of time...

100. TuringTest Group Title

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

101. Inopeki Group Title

x^7?

102. TuringTest Group Title

good :) more time saved...

103. Inopeki Group Title

:D actually i didnt know that

104. TuringTest Group Title

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 Group Title

Yeah, i assumed that :)

106. TuringTest Group Title

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 Group Title

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 Group Title

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 Group Title

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

110. Inopeki Group Title

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

111. TuringTest Group Title

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

112. Inopeki Group Title

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 Group Title

good so far :)

114. Inopeki Group Title

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

115. Inopeki Group Title

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 Group Title

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

117. Inopeki Group Title

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

118. TuringTest Group Title

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

119. Inopeki Group Title

Oh right lol

120. TuringTest Group Title

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

121. Inopeki Group Title

What?

122. TuringTest Group Title

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

123. Inopeki Group Title

abcd?

124. TuringTest Group Title

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 Group Title

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

126. Inopeki Group Title

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

127. TuringTest Group Title

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 Group Title

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

129. TuringTest Group Title

nice! you can simplify the middle terms

130. Inopeki Group Title

Oh right,x^2+5x+6

131. TuringTest Group Title

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

132. Inopeki Group Title

5x+2x+15x-5?

133. Inopeki Group Title

6x*

134. TuringTest Group Title

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

135. TuringTest Group Title

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

136. Inopeki Group Title

-1?

137. TuringTest Group Title

times what?

138. Inopeki Group Title

2x, maybe thats -2x..

139. TuringTest Group Title

there ya go ;-)

140. TuringTest Group Title

so it should be...?

141. Inopeki Group Title

6x+-2x+15x-5

142. TuringTest Group Title

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

143. Inopeki Group Title

29x-5?

144. TuringTest Group Title

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

145. Inopeki Group Title

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

146. TuringTest Group Title

yes, and now simplify

147. Inopeki Group Title

4x^2+15x-5?

148. TuringTest Group Title

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

149. Inopeki Group Title

so 6x^2+13x-5?

150. TuringTest Group Title

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

151. Inopeki Group Title

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

152. TuringTest Group Title

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

153. Inopeki Group Title

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

154. Inopeki Group Title

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

155. TuringTest Group Title

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 Group Title

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

157. TuringTest Group Title

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

158. Inopeki Group Title

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

159. TuringTest Group Title

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

160. Inopeki Group Title

I cant remember :(

161. TuringTest Group Title

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

162. Inopeki Group Title

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

163. TuringTest Group Title

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

164. TuringTest Group Title

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

165. Inopeki Group Title

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

166. TuringTest Group Title

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

167. Inopeki Group Title

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

168. TuringTest Group Title

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

169. Inopeki Group Title

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

170. TuringTest Group Title

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

171. TuringTest Group Title

no*

172. TuringTest Group Title

can you simplify the last step?

173. Inopeki Group Title

Yeah, i need to sleep.

174. TuringTest Group Title

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 Group Title

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