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  • By Remainder Theorem find the remainder, when p(x) is divided by g(x) , where (i)p(x)=x^{3}-2x^{2}-4x-1; g(x)=x+1 (ii)p(x)=x^{3}-3x^{2}+4x+50; g(x)=x-3 (iii)p(x)=4x^{3}-12x^{2}+14x-3; g(x)=2x-1 (iv)p(x)=x^{3}-6x^{2}+2x-4; g(x)=1-\frac{3}{2}x

By Remainder Theorem find the remainder, when p(x)  is divided by g(x) , where

(i)p(x)=x^{3}-2x^{2}-4x-1; g(x)=x+1

(ii)p(x)=x^{3}-3x^{2}+4x+50; g(x)=x-3

(iii)p(x)=4x^{3}-12x^{2}+14x-3; g(x)=2x-1

(iv)p(x)=x^{3}-6x^{2}+2x-4; g(x)=1-\frac{3}{2}x

           

Answers (1)

(i) 0

Solution:-  According to remainder theorem when p(x)  is divided by (x+a)  then the remainder is p(-a) .

p(x)=x^{3}-2x^{2}-4x-1; g(x)=x+1

So, when p(x)  is divided by g(x)  then remainder will be p(-1) .

\\p(-1)=(-1)^{3}-2(-1)^{2}-4(-1)-1\\ =-1-2+4-1\\ =-4+4\\ =0

Hence the remainder is zero

(ii)62

Solution:-  According to remainder theorem when p(x)  is divided by (x+a)  then the remainder is p(-a) .

p(x)=x^{3}-3x^{2}+4x+50; g(x)=x-3           
So, when p(x)  is divided by g(x)  then remainder will be p(3) .
\\p(3)=3^{3}-3(3)^{2}+4(3)+50\\ p(3)=27-27+12+50\\ p(3)=62

Hence the remainder is 62.

(iii) \frac{3}{2}

Solution:-  According to remainder theorem when p(x)  is divided by (x+a)  then the remainder is p(-a) .

p(x)=4x^{3}-12x^{2}+14x-3; g(x)=2x-1           
So, when p(x)  is divided by g(x)  then remainder will bep\left (\frac{1}{2} \right )
\\p\left(\frac{1}{2} \right )=4\left (\frac{1}{2} \right )^{3}-12 \left (\frac{1}{2} \right )^{2}+14 \left (\frac{1}{2} \right )-3\\ =4 \times \frac{1}{8}-12\times \frac{1}{4}+7-3\\ =\frac{1}{2}-3+7-3\\ =\frac{1}{2}+1\\ =\frac{1+2}{2}=\frac{3}{2}

Hence the remainder is \frac{3}{2}

(iv) -\frac{136}{27}

Solution:-  According to remainder theorem when p(x)  is divided by (x+a)  then the remainder is p(-a) .

p(x)=x^{3}-6x^{2}+2x-4; g(x)=1-\frac{3}{2}x           
So, when p(x)  is divided by g(x)  then remainder will bep\left (\frac{2}{3} \right )

\\p\left (\frac{2}{3} \right )=\left (\frac{2}{3} \right )^{3}-6\left (\frac{2}{3} \right )^{2}+2\left (\frac{2}{3} \right )-4\\ =\frac{8}{27}-6\left ( \frac{4}{9} \right )+\frac{4}{3}-4\\ =\frac{8-72+36-108}{27}\\ =-\frac{136}{27}

Hence the remainder is -\frac{136}{27}.

           

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