when a polynomial is divided by x-1,x+1andx+2 it leaves the reminders -1,1and2 respectively.Find the reminder when it is divided by x^3+2x^2-x-2.
Answers
let the polynomial be F(x)
APQ: by applying remainder theorem,
we get the following equations
then, we get
and finally we get
so, we will factorise the polynomial given
so, the polynomial f(x) will be such that
where, q(x) is the quotient and r(x) is the remainder. since we know,
where deg(r(x)) means degree of r(x).
so, let use assume,
returning back to our initial equation we get
so, substituting the values woven in the question I.e. x=1,-1,2 we get the following equations
which simplifies to be
Eq... 1
similarly we get the following equations
this again simplifies to be
Eq... 2
again repeating we get
this again simplifies to be
Eq... 3
solving these equations, we will get the value of a, b and c.
adding Eq 1 and 2
or
or
so, then, we will be able to get the value of b in Eq 1 given
so since c= --a
substituting b= --1 and c= --a in Eq 3 we get
so,
c= --a=0
substituting the values of a, b and c, we will get the r(x)
which simplifies to be
HOPE THIS HELPS YOU. ALTHOUGH YOU COULD HAVE IDENTIFIED THE PATTERN BY THE VALUES GIVEN IN THE QUESTION, YOU SHOUKD ALWAYS PREFER WRITING STEP-WISE IN EXAMS.
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