What must be subtracted from x^3-6x^2-15x+80 so that the result is exactly divisible by x^2+x-12?find using factor theorem.

Asked by abinash.gupta003 | 26th Jul, 2016, 06:39: PM

Expert Answer:

begin mathsize 16px style By space the space division space algorithm comma space when space straight p left parenthesis straight x right parenthesis space equals space straight x cubed minus 6 straight x squared minus 15 straight x space plus space 80 space is space divided space by space straight q left parenthesis straight x right parenthesis equals straight x squared plus straight x minus 12 comma space the space remainder space is space straight a space linear space expression.
So comma space let space straight r left parenthesis straight x right parenthesis equals ax plus straight b space be space subtracted space from space straight p left parenthesis straight x right parenthesis space so space that space the space result space is space divisible space by space straight q left parenthesis straight x right parenthesis.
Let space straight f left parenthesis straight x right parenthesis equals straight p left parenthesis straight x right parenthesis minus straight r left parenthesis straight x right parenthesis equals straight p left parenthesis straight x right parenthesis minus left parenthesis ax plus straight b right parenthesis
or comma space straight f left parenthesis straight x right parenthesis equals space straight x cubed minus 6 straight x squared minus 15 straight x plus 80 space minus space left parenthesis ax plus straight b right parenthesis
or comma space straight f left parenthesis straight x right parenthesis equals space straight x cubed minus 6 straight x squared minus straight x left parenthesis straight a plus 15 right parenthesis plus 80 minus straight b
We space have comma
straight q left parenthesis straight x right parenthesis equals straight x squared plus straight x minus 12 equals left parenthesis straight x plus 4 right parenthesis left parenthesis straight x minus 3 right parenthesis
Clearly comma space straight x plus 4 space and space straight x minus 3 space are space factors space of space straight q left parenthesis straight x right parenthesis.
Therefore comma space straight f left parenthesis straight x right parenthesis space will space be space divisible space by space straight q left parenthesis straight x right parenthesis space if space straight x plus 4 space and space straight x minus 3 space are space factors space of space straight f left parenthesis straight x right parenthesis.
that space is comma space straight f left parenthesis negative 4 right parenthesis equals 0 space and space straight f left parenthesis 3 right parenthesis equals 0 space space space left square bracket Since space straight x plus 4 equals 0 rightwards double arrow straight x equals negative 4 space and space straight x minus 3 equals 0 rightwards double arrow straight x equals 3 right square bracket
rightwards double arrow left parenthesis negative 4 right parenthesis cubed minus 6 left parenthesis negative 4 right parenthesis squared plus 4 left parenthesis straight a plus 15 right parenthesis plus 80 minus straight b equals 0 space and space left parenthesis 3 right parenthesis cubed minus 6 left parenthesis 3 right parenthesis squared minus 3 left parenthesis straight a plus 15 right parenthesis plus 80 minus straight b equals 0
rightwards double arrow negative 64 minus 96 plus 4 straight a plus 60 plus 80 minus straight b equals 0 space and space 27 minus 54 minus 3 straight a minus 45 plus 80 minus straight b equals 0
rightwards double arrow 4 straight a minus straight b equals 20 space and space minus 3 straight a minus straight b equals negative 8
rightwards double arrow Solving space for space straight a space and space straight b comma space we space get space straight a equals 4 space and space straight b equals negative 4.
Substituting space the space value space of space straight a space and space straight b space in space straight r left parenthesis straight x right parenthesis equals ax plus straight b comma space we space get
straight r left parenthesis straight x right parenthesis equals 4 straight x minus 4
Hence comma space straight p left parenthesis straight x right parenthesis space is space divisible space by space straight q left parenthesis straight x right parenthesis comma space if space straight r left parenthesis straight x right parenthesis equals 4 straight x minus 4 space is space subtracted space from space it. end style

Answered by Rebecca Fernandes | 27th Jul, 2016, 09:31: AM

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