Question

to prove remainder theorem

Answer 1

Hey!!______ here is your solution.⤵⤵⤵⤵

Suppose p(x)p(x) is a non-zero polynomial with real coefficients and cc is a real number.

Dividingp(x)p(x) by a polynomialq(x),q(x),we get a polynomials(x)s(x)as quotient and a polynomial r(x)r(x)as remainder where 0≤degr(x)<degq(x).0≤deg⁡r(x)<deg⁡q(x).

Here we getp(x)=q(x).s(x)+r(x).p(x)=q(x).s(x)+r(x).

Therefore by dividingp(x)p(x) byx−cx−c we get a polynomial s(x)s(x)as quotient and a constantrr as remainder and we get the relation that

p(x)=q(x).(x−c)+rp(x)=q(x).(x−c)+r ……….. (1)

From the relation, by putting the value ccforxxin (1),we get

p(c)=r.p(c)=r.

Thus , the remainder of p(x)p(x) when it is divided by(x−c)(x−c)is the number p(c).p(c).This result is called remainder theorem.

From the same result we get, the remainder is zero if and only if p(c)p(c)is zero.

This result can be restated as follows:

x−cx−cis a factor of p(x)p(x)if and only if p(c)=0.p(c)=0.

This result is called Factor theorem.

These simple results are very useful in factorization of polynomials and evaluating remainders on division by linear polynomials.

Example:

Suppose we want to find the remainder of 2x7−4x3+8x−122x7−4x3+8x−12when it is divided by 2x+42x+4

Note that the quotient and remainder will not affected by dividing by a non-zero number both dividend and divisor.

Hence we find the remainder of p(x)=x7−2x3+4x−6p(x)=x7−2x3+4x−6when it is divided by x+2x+2

By applying remainder theorem we get the remainder is p(−2)=−128+16−8−6=−126p(−2)=−128+16−8−6=−126

sincex+2=x−(−2).x+2=x−(−2).

Thus required remainder is −126−126✔✔✔

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Hope it helps:-)