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Multiplication of Polynomial

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donald trump
19/03/2017 at 15:22
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It is given (r-s) = 4 and (t-s) = 9

Find the value of r2+s2+t2-rs-st-rt.

Multiplication of Polynomial

  • ...
    Dung Trần Thùy 19/03/2017 at 22:19

    \(\left\{{}\begin{matrix}r-s=4\\t-s=9\end{matrix}\right.\Rightarrow\left(t-s\right)-\left(r-s\right)=t-r=9-4=5\)

    We have :

    \(r^2+s^2+t^2-rs-st-rt\)

    \(=\dfrac{2r^2+2s^2+2t^2-2rs-2st-2rt}{2}\)

    \(=\dfrac{\left(r^2+s^2-2rs\right)+\left(r^2+t^2-2rt\right)+\left(t^2+s^2-2ts\right)}{2}\)

    \(=\dfrac{\left(r-s\right)^2+\left(t-r\right)^2+\left(t-s\right)^2}{2}\)

    \(=\dfrac{4^2+5^2+9^2}{2}=\dfrac{122}{2}=61\)

    Ans : 61

    donald trump selected this answer.
  • ...
    FA KAKALOTS 06/02/2018 at 12:36

    {r−s=4t−s=9⇒(t−s)−(r−s)=t−r=9−4=5

    We have :

    r2+s2+t2−rs−st−rt

    =2r2+2s2+2t2−2rs−2st−2rt2

    =(r2+s2−2rs)+(r2+t2−2rt)+(t2+s2−2ts)2

    =(r−s)2+(t−r)2+(t−s)22

    =42+52+922=1222=61

    Ans : 61


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donald trump
19/03/2017 at 15:20
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It is given (a+b)2 = 1, (a-b)2 = 4 and (a+b)2 = 64

Find the value of \(\dfrac{a}{b}+\dfrac{b}{a}\)

Multiplication of Polynomial

  • ...
    Dung Trần Thùy 19/03/2017 at 22:08

    You must check ur problem ok


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donald trump
19/03/2017 at 15:13
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Evaluate (6+1)(62+1)(64+1)(68+1)(616+1).

Multiplication of Polynomial

  • ...
    Dung Trần Thùy 19/03/2017 at 22:10

    Let \(A=\left(6+1\right)\left(6^2+1\right)...\left(6^{16}+1\right)\)

    ​ ​\(\Rightarrow A=\dfrac{1}{5}.5.\left(6+1\right)\left(6^2+1\right)...\left(6^{16}+1\right)\)

    ​ ​\(\Rightarrow A=\dfrac{1}{5}.\left(6-1\right).\left(6+1\right)\left(6^2+1\right)...\left(6^{16}+1\right)\)

    \(\Rightarrow A=\dfrac{1}{5}.\left[\left(6-1\right)\left(6+1\right)\right]\left(6^2+1\right)...\left(6^{16}+1\right)\)

    \(\Rightarrow A=\dfrac{1}{5}.\left(6^2-1\right)\left(6^2+1\right)\left(6^4+1\right)...\left(6^{16}+1\right)\)

    \(...\)

    \(\Rightarrow A=\dfrac{1}{5}.\left(6^{32}-1\right)=\dfrac{6^{32}-1}{5}\)

    ok

    donald trump selected this answer.
  • ...
    FA KAKALOTS 06/02/2018 at 12:36

    Let A=(6+1)(62+1)...(616+1)

    ​ ​⇒A=15.5.(6+1)(62+1)...(616+1)

    ​ ​⇒A=15.(6−1).(6+1)(62+1)...(616+1)

    ⇒A=15.[(6−1)(6+1)](62+1)...(616+1)

    ⇒A=15.(62−1)(62+1)(64+1)...(616+1)

    ...

    ⇒A=15.(632−1)=632−15oe


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donald trump
19/03/2017 at 15:11
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It is given x2 - 5x + 1 = 0. Find the value of x2 + \(\dfrac{1}{x^2}\)

Multiplication of Polynomial

  • ...
    Dung Trần Thùy 19/03/2017 at 22:26

    We have :

    \(x^2-5x+1=0\)

    \(\Rightarrow x^2+1=5x\) ( 1 )

    \(x^2+\dfrac{1}{x^2}=\left(x^2+\dfrac{1}{x^2}+2.\dfrac{1}{x}.x\right)-2.\dfrac{1}{x}.x=\left(x+\dfrac{1}{x}\right)^2-2\)

                    \(=\left(\dfrac{x^2+1}{x}\right)^2-2\) ( 2 )

    ( 1 )( 2 )\(\Rightarrow x^2+\dfrac{1}{x^2}=\left(\dfrac{5x}{x}\right)^2-2=25-2=23\)

    Ans : 23.

    donald trump selected this answer.
  • ...
    ¤« 06/04/2018 at 13:06

    We have :

    x2−5x+1=0

    ⇒x2+1=5x

     ( 1 )

    x2+1x2=(x2+1x2+2.1x.x)−2.1x.x=(x+1x)2−2

                    =(x2+1x)2−2

     ( 2 )

    ( 1 )( 2 )⇒x2+1x2=(5xx)2−2=25−2=23

    Ans : 23.


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donald trump
19/03/2017 at 15:10
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It is given m + n = 5 and mn = 6. Find the value of m3 + n3

Multiplication of Polynomial

  • ...
    FA KAKALOTS 06/02/2018 at 12:37

    (m+n)2=52

    ⇒m2+n2+2mn=25

    ⇒m2+n2=25−2mn=25−2.6=25−12=13

    ⇒m3+n3=(m+n)(m2+n2−mn)=5.(13−6)=35

    Ans : 35.

  • ...
    Dung Trần Thùy 19/03/2017 at 22:27

    \(\left(m+n\right)^2=5^2\)

    \(\Rightarrow m^2+n^2+2mn=25\)

    \(\Rightarrow m^2+n^2=25-2mn=25-2.6=25-12=13\)

    \(\Rightarrow m^3+n^3=\left(m+n\right)\left(m^2+n^2-mn\right)=5.\left(13-6\right)=35\)

    Ans : 35.


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taylor swift
19/03/2017 at 15:07
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The side of a square is S cm.

The length and width of a rectangle is (S+3) cm and (S-3) cm respectively. Find the sum of the areas of the square and rectangle

Multiplication of Polynomial

  • ...
    Dung Trần Thùy 19/03/2017 at 22:29

    The area of the square is S2.

    The area of the rectangle is \((S+3)(S-3)=S^2 - 9\)

    So, the sum of the areas of 2 fingures is \(S^2+\left(S^2-9\right)=2S^2-9\).

    taylor swift selected this answer.
  • ...
    FA KAKALOTS 06/02/2018 at 12:37

    The area of the square is S2.

    The area of the rectangle is (S+3)(S−3)=S2−9

    So, the sum of the areas of 2 fingures is S2+(S2−9)=2S2−9

    .


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taylor swift
19/03/2017 at 15:04
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Use a simple method to compute 

12 - 22 + 32 - 42 + ... + 19992 - 20002

Multiplication of Polynomial

  • ...
    Dung Trần Thùy 19/03/2017 at 22:14

    \(=\left(1^2-2^2\right)+\left(3^2-4^2\right)+\left(5^2-6^2\right)+\left(1999^2-2000^2\right)\)

    \(=\left(-3\right)+\left(-7\right)+\left(-11\right)+...+\left(-3999\right)\)

    The number of  terms is \(\dfrac{\left(-3\right)-\left(-3999\right)}{4}+1=1000\) ( terms )

    So the value of this expression is \(\dfrac{1000.\left[\left(-3\right)+\left(-3999\right)\right]}{2}=-2001000\)

    taylor swift selected this answer.

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taylor swift
19/03/2017 at 14:59
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(a+22) is a square number: (a-23) is another square number . Find the value of a

Multiplication of Polynomial

  • ...
    FA KAKALOTS 06/02/2018 at 12:37

    Let m2 and n2 be a + 22 and a - 23 respectively. ( m > n > 0)

    So, we have m2−n2=(a+22)−(a−23)

    ⇒(m−n)(m+n)=45=1.45=3.15=5.9

    ⋅

     If m - n =  1 and m + n = 45; then m = 23 ; n = 22; so a=m2−22=n2+23=507

    ⋅

     If m - n = 3 and m + n = 15; then m = 9 and n = 6, so a = 59.

    ⋅

     If m - n = 5 and m + n = 9; then m = 7 and n = 2; so a = 27.

    Ans : a∈{27;59;507}.

  • ...
    Dung Trần Thùy 19/03/2017 at 22:39

    Let m2 and n2 be a + 22 and a - 23 respectively. ( m > n > 0)

    So, we have \(m^2-n^2=\left(a+22\right)-\left(a-23\right)\)

    \(\Rightarrow\left(m-n\right)\left(m+n\right)=45=1.45=3.15=5.9\)

    \(\cdot\) If m - n =  1 and m + n = 45; then m = 23 ; n = 22; so \(a=m^2-22=n^2+23=507\)

    \(\cdot\) If m - n = 3 and m + n = 15; then m = 9 and n = 6, so a = 59.

    \(\cdot\) If m - n = 5 and m + n = 9; then m = 7 and n = 2; so a = 27.

    Ans : \(a\in\left\{27;59;507\right\}.\)


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taylor swift
19/03/2017 at 14:58
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Use a simple method to compute

\(\dfrac{\left(2002\right)^2}{\left(2001\right)^2+\left(2003\right)^2-2}\)

Multiplication of Polynomial

  • ...
    FA KAKALOTS 06/02/2018 at 12:38

    2002^2/2001^2+2003^2-2

    2002^2/ (2002-1)^2+(2002+1)^2-2

     Have: a^2 - ( a-1)^2 = a+(a-1)

    2002^2/ ( 2002^2-4003) + ( 2002^2 +4005 ) -2

    2002^2/2002^2-4003+2002^2+4005-2

    2002^2/2002^2+2002^2-4003+4005-2

    =2002^2/2002^2+2002^2

    = 2002^2/ 2002^2.2 

    = 1/2

    Answer: 1/2

  • ...
    In the name of love 20/03/2017 at 15:41

    2002^2/2001^2+2003^2-2

    2002^2/ (2002-1)^2+(2002+1)^2-2

     Have: a^2 - ( a-1)^2 = a+(a-1)

    2002^2/ ( 2002^2-4003) + ( 2002^2 +4005 ) -2

    2002^2/2002^2-4003+2002^2+4005-2

    2002^2/2002^2+2002^2-4003+4005-2

    =2002^2/2002^2+2002^2

    = 2002^2/ 2002^2.2 

    = 1/2

    Answer: 1/2


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taylor swift
19/03/2017 at 14:55
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Observe the pattern for multiplication of polynomials

(x-1)(x+1) = x2-1

(x-1)(x2+x+1) = x3-1

(x-1)(x3+x2+x+1) = x4-1

The result of (x-1)(xn+xn-1+xn-2+...+1) is

Multiplication of Polynomial

  • ...
    Dung Trần Thùy 19/03/2017 at 22:40

    Ans :  \(x^{n+1}-1\)


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taylor swift
19/03/2017 at 14:51
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If a2 + a = 0, then a2001 + a2000 + 7 = 

Multiplication of Polynomial

  • ...
    FA KAKALOTS 06/02/2018 at 12:38

    a2+a=a(a+1)=0

    ⇔[a=0a=−1

    ⇔[a2001+a2000+7=0+0+7=7a2001+a2000+7=(−1)+1+7=7

    Ans : 7.

  • ...
    Dung Trần Thùy 19/03/2017 at 22:41

    \(a^2+a=a\left(a+1\right)=0\)

    \(\Leftrightarrow\left[{}\begin{matrix}a=0\\a=-1\end{matrix}\right.\)

    \(\Leftrightarrow\left[{}\begin{matrix}a^{2001}+a^{2000}+7=0+0+7=7\\a^{2001}+a^{2000}+7=\left(-1\right)+1+7=7\end{matrix}\right.\)

    Ans : 7.


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taylor swift
19/03/2017 at 14:48
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Evaluate 

\(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)...\left(1-\dfrac{1}{11^2}\right)\left(1-\dfrac{1}{12^2}\right)\)

Multiplication of Polynomial

  • ...
    FA KAKALOTS 06/02/2018 at 12:38

    (1−122)(1−132)...(1−1112)(1−1122)

    =(1−12)(1+12)(1−13)(1+13)...(1−111)(1+111)(1−112)(1+112)

    =(12.32)(23.43)....(1011.1211)(1112.1312)

    =(12.23...1011.1112)(32.43...1211.1312)

    =112.132=1324

  • ...
    Phan Thanh Tinh Coordinator 24/04/2017 at 22:52

    \(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)...\left(1-\dfrac{1}{11^2}\right)\left(1-\dfrac{1}{12^2}\right)\)

    \(=\left(1-\dfrac{1}{2}\right)\left(1+\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1+\dfrac{1}{3}\right)...\left(1-\dfrac{1}{11}\right)\left(1+\dfrac{1}{11}\right)\left(1-\dfrac{1}{12}\right)\left(1+\dfrac{1}{12}\right)\)

    \(=\left(\dfrac{1}{2}.\dfrac{3}{2}\right)\left(\dfrac{2}{3}.\dfrac{4}{3}\right)....\left(\dfrac{10}{11}.\dfrac{12}{11}\right)\left(\dfrac{11}{12}.\dfrac{13}{12}\right)\)

    \(=\left(\dfrac{1}{2}.\dfrac{2}{3}...\dfrac{10}{11}.\dfrac{11}{12}\right)\left(\dfrac{3}{2}.\dfrac{4}{3}...\dfrac{12}{11}.\dfrac{13}{12}\right)\)

    \(=\dfrac{1}{12}.\dfrac{13}{2}=\dfrac{13}{24}\)


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taylor swift
19/03/2017 at 14:44
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The result of (234567)2 - 234557 x 234577 is  ... when we use a simple method to compute

Multiplication of Polynomial

  • ...
    FA KAKALOTS 06/02/2018 at 12:38

    (1−122)(1−132)...(1−1112)(1−1122)

    =(1−12)(1+12)(1−13)(1+13)...(1−111)(1+111)(1−112)(1+112)

    =(12.32)(23.43)....(1011.1211)(1112.1312)

    =(12.23...1011.1112)(32.43...1211.1312)

    =112.132=1324

  • ...
    Math You Like 04/09/2017 at 15:27

    We have: 2345672 - 234557 . 234577

    = 2345672 - (234567 - 10) . (234567 + 10)

    =2345672 - (2345672 - 102)

    =102 =100

  • ...
    Phan Thanh Tinh Coordinator 24/04/2017 at 22:55

    We have :

    2345672 - 234557.234577

    = 2345672 - (234567 - 10)(234567 + 10) 

    = 2345672 - (2345672 - 102)

    = 102 = 100


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taylor swift
19/03/2017 at 12:59
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It is given n is a positive integer . The units digit of n2 is 9 and the units digit of (n+1)2 is 6. The units digit of (n-1)2 is

Multiplication of Polynomial

  • ...
    FA KAKALOTS 06/02/2018 at 12:39

    The units digit of n2 is 9,so the unit digit of n can be 3 or 7 (because 32 = 9 ; 72 = 49)

    The units digit of (n + 1)2 is 6,so the unit digit of n + 1 can be 4 or 6

    We see 3 + 1 = 4,so the units digit of n is 3 and the units digit of n - 1 is 2.Hence the units digit of (n - 1)2 is 4

  • ...
    »ﻲ2004#ﻲ« 29/03/2017 at 06:13

    The units digit of n2 is 9,so the unit digit of n can be 3 or 7 (because 32 = 9 ; 72 = 49)

    The units digit of (n + 1)2 is 6,so the unit digit of n + 1 can be 4 or 6

    We see 3 + 1 = 4,so the units digit of n is 3 and the units digit of n - 1 is 2.Hence the units digit of (n - 1)2 is 4

  • ...
    Phan Thanh Tinh Coordinator 23/03/2017 at 17:42

    The units digit of n2 is 9,so the unit digit of n can be 3 or 7 (because 32 = 9 ; 72 = 49)

    The units digit of (n + 1)2 is 6,so the unit digit of n + 1 can be 4 or 6

    We see 3 + 1 = 4,so the units digit of n is 3 and the units digit of n - 1 is 2.Hence the units digit of (n - 1)2 is 4


...
mark zuckerberg
19/03/2017 at 00:32
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Use a simple method to compute S = (2+1)(22+1)(24+1)...(220+1).

Multiplication of Polynomial

  • ...
    Phan Thanh Tinh Coordinator 23/03/2017 at 18:46

    S = (2 + 1)(22 + 1)(24 + 1)...(220 + 1)

       = (2 - 1)(2 + 1)(22 + 1)(24 + 1)...(220 + 1)

       = (22 - 1)(22 + 1)(24 + 1)...(220 + 1)

       = (24 - 1)(24 + 1)...(220 + 1)

    ..........................................................

       = (220 - 1)(220 + 1)

       = 240 - 1

    Selected by MathYouLike
  • ...
    FA KAKALOTS 06/02/2018 at 12:39

    S = (2 + 1)(22 + 1)(24 + 1)...(220 + 1)

       = (2 - 1)(2 + 1)(22 + 1)(24 + 1)...(220 + 1)

       = (22 - 1)(22 + 1)(24 + 1)...(220 + 1)

       = (24 - 1)(24 + 1)...(220 + 1)

    ..........................................................

       = (220 - 1)(220 + 1)

       = 240 - 1

  • ...
    »ﻲ2004#ﻲ« 29/03/2017 at 06:12

    S = (2 + 1)(22 + 1)(24 + 1)...(220 + 1)

       = (2 - 1)(2 + 1)(22 + 1)(24 + 1)...(220 + 1)

       = (22 - 1)(22 + 1)(24 + 1)...(220 + 1)

       = (24 - 1)(24 + 1)...(220 + 1)

       = (220 - 1)(220 + 1)

       = 240 - 1


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mark zuckerberg
19/03/2017 at 00:31
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It is given a - b = 4. Find the value of a3 - b3 - 12ab.

Multiplication of Polynomial


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mark zuckerberg
19/03/2017 at 00:29
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It is given \(m+\dfrac{1}{m}=4.\)Find the value of \(m^4+\dfrac{1}{m^4}\)

Multiplication of Polynomial

  • ...
    FA KAKALOTS 06/02/2018 at 12:39

    We have :

    m4+1m4=m4+2+1m4−2=(m2+1m2)2−2

    =(m2+2+1m2−2)2−2=[(m+1m)2−2]2−2

    =(42−2)2−2=142−2=194

  • ...
    Phan Thanh Tinh Coordinator 23/05/2017 at 09:10

    We have :

    \(m^4+\dfrac{1}{m^4}=m^4+2+\dfrac{1}{m^4}-2=\left(m^2+\dfrac{1}{m^2}\right)^2-2\)

    \(=\left(m^2+2+\dfrac{1}{m^2}-2\right)^2-2=\left[\left(m+\dfrac{1}{m}\right)^2-2\right]^2-2\)

    \(=\left(4^2-2\right)^2-2=14^2-2=194\)


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