Question

If the polynomialx^4+2x^3+8x^2+12x+18 is divided by another polynomial x^2+5,the remainder comes out to be px+q,find the value of p & q

Answer 1

Here the polynomial $f(x)=x^4+2x^3+8x^2+12x+18$ is divided by the polynomial $x^2+5,$ so we have to express $f(x)$ in the form $g(x)\cdot(x^2+5)+px+q$ where $g(x)$ is for some second degree polynomial (the quotient) and $px+q$ is the remainder.

$\longrightarrow f(x)=g(x)\cdot(x^2+5)+(px+q)\quad\quad\dots(1)$

Then,

$\longrightarrow f(x)=x^4+2x^3+8x^2+12x+18$

Taking $8x^2=5x^2+3x^2,\ 12x=10x+2x$ and $18=15+3,$

$\longrightarrow f(x)=x^4+2x^3+5x^2+3x^2+10x+2x+15+3$

Or,

$\longrightarrow f(x)=x^4+5x^2+2x^3+10x+3x^2+15+2x+3$

$\longrightarrow f(x)=x^2(x^2+5)+2x(x^2+5)+3(x^2+5)+(2x+3)$

$\longrightarrow f(x)=(x^2+2x+3)(x^2+5)+(2x+3)\quad\quad\dots(2)$

From (1) and (2),

$\longrightarrow px+q=2x+3$

Thus we get,

$\longrightarrow\underline{\underline{p=2}}$

$\longrightarrow\underline{\underline{q=3}}$