find the values of p and q if the polynomial g(x)=x^2-3x-10 is a factor of the polynomial f(x)=x^3+px^2+qx+20.

Asked by bdroy_2014 | 5th Feb, 2018, 10:55: PM

Expert Answer:

begin mathsize 16px style straight g left parenthesis straight x right parenthesis equals straight x squared minus 3 straight x minus 10
equals straight x squared minus 5 straight x plus 2 straight x minus 10
equals straight x left parenthesis straight x minus 5 right parenthesis plus 2 left parenthesis straight x minus 5 right parenthesis
equals left parenthesis straight x minus 5 right parenthesis left parenthesis straight x plus 2 right parenthesis
rightwards double arrow left parenthesis straight x minus 5 right parenthesis space and space left parenthesis straight x plus 2 right parenthesis space are space factors space of space straight f left parenthesis straight x right parenthesis equals straight x cubed plus px squared plus qx plus 20
Now comma space left parenthesis straight x minus 5 right parenthesis space is space straight a space factor space of space straight f left parenthesis straight x right parenthesis equals straight x cubed plus px squared plus qx plus 20
rightwards double arrow left parenthesis 5 right parenthesis cubed plus straight p left parenthesis 5 right parenthesis squared plus straight q left parenthesis 5 right parenthesis plus 20 equals 0
rightwards double arrow 125 plus 25 straight p plus 5 straight q plus 20 equals 0
rightwards double arrow 25 straight p plus 5 straight q equals negative 145
rightwards double arrow 5 straight p plus straight q equals negative 29 space space space space space space.... left parenthesis straight i right parenthesis
And comma space left parenthesis straight x plus 2 right parenthesis space is space straight a space factor space of space straight f left parenthesis straight x right parenthesis equals straight x cubed plus px squared plus qx plus 20
rightwards double arrow left parenthesis negative 2 right parenthesis cubed plus straight p left parenthesis negative 2 right parenthesis squared plus straight q left parenthesis negative 2 right parenthesis plus 20 equals 0
rightwards double arrow negative 8 plus 4 straight p minus 2 straight q plus 20 equals 0
rightwards double arrow 4 straight p minus 2 straight q equals negative 12
rightwards double arrow 2 straight p minus straight q equals negative 6 space space space space space space.... left parenthesis ii right parenthesis
Adding space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma
7 straight p equals negative 35
rightwards double arrow straight p equals negative 5
rightwards double arrow straight q equals negative 29 minus 5 left parenthesis negative 5 right parenthesis equals negative 29 plus 25 equals negative 4 end style

Answered by Rashmi Khot | 6th Feb, 2018, 09:50: AM