Question

4. The polynomial p(x) = x* - 2x + 3x? - ax + b when divided by (x - 1) and
(x + 1) leaves the remainders 5 and 19 respectively. Find the values of a
and b. Hence, find the remainder when p(x) is divided by (x - 2).​

Answer 1

Answer:

There is theorem known as “Polynomial Remainder Theorem” or “ Bezout’s Theorem”. It is Stated as -

A Polynomial f(x) if divided by a linear polynomial (x-a) leaves remainder which equals f(a).

So , getting back to our question -

f(x)=x4−2x3+3x2−ax+b

So , when it is divided by (x−1) it’ll leave a remainder = f(1) = 5 (Given).

f(1)=14−2×13+3×12−a×1+b=5

=>1−2+3−a+b=5

=>a−b=(−3)….Eqn(1)

Now , Similarly -

f(−1)=(−1)4−2×(−1)3+3×(−1)2−a×(−1)+b=19

=>1+2+3+a+b=19

=>a+b=13….Eqn(2)

Now , adding equations (1) and (2) , We’ll get -

(a+b)+(a−b)=(−3)+13

=>2a=10=>a=5

So , (a+b)=13 implies b=8

Hence , Values of a and b are 5 and 8 respectively.

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