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Question Number 205018 by BaliramKumar last updated on 06/Mar/24
For what value of  ′k′ can be expression x^3  + kx^2  −7x +6              be resolved into three linear factors?  (a) 0           (b) 1             (c) 2              (d) 3
Forwhatvalueofkcanbeexpressionx3+kx27x+6beresolvedintothreelinearfactors?(a)0(b)1(c)2(d)3
Commented by Rasheed.Sindhi last updated on 06/Mar/24
(a) k=0
(a)k=0
Commented by mr W last updated on 06/Mar/24
question is not clear. in complex  numbers with any real value of k  the expression can always be resolved  into three linear factors. in real  numbers the expression can be  resolved into three linear factors  if k_1 <k<k_2 .
questionisnotclear.incomplexnumberswithanyrealvalueofktheexpressioncanalwaysberesolvedintothreelinearfactors.inrealnumberstheexpressioncanberesolvedintothreelinearfactorsifk1<k<k2.
Answered by Rasheed.Sindhi last updated on 06/Mar/24
(a)k=0  x^3  + kx^2  −7x +6⇒x^3   −7x +6  x^3  −1 −7x +7  (x−1)(x^2 +x+1)−7(x−1)  (x−1)(x^2 +x−6)  (x−1)(x+3)(x−2)  For k=1,2,3 the expression has  no linear factor.
(a)k=0x3+kx27x+6x37x+6x317x+7(x1)(x2+x+1)7(x1)(x1)(x2+x6)(x1)(x+3)(x2)Fork=1,2,3theexpressionhasnolinearfactor.
Commented by mr W last updated on 06/Mar/24
linear factors (x+a)(x+b)(x+c)  don′t request that a, b, c should be  integer numbers. they may be real  numbers or even complex numbers.
linearfactors(x+a)(x+b)(x+c)dontrequestthata,b,cshouldbeintegernumbers.theymayberealnumbersorevencomplexnumbers.
Commented by Rasheed.Sindhi last updated on 06/Mar/24
Sir, I considered only numbers given  as options:  (a) k=0    (b) k=1  (c) k=2   (d) k=3
Sir,Iconsideredonlynumbersgivenasoptions:(a)k=0(b)k=1(c)k=2(d)k=3
Commented by mr W last updated on 06/Mar/24
yes. from these 4 given options only  k=0 is suitable.
yes.fromthese4givenoptionsonlyk=0issuitable.

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