Give an example of polynomials f(x), g(x), q(x) and r(x) satisfying f(x) = g(x), q(x) + r(x), where degree r(x) = 0.
Using division algorithm, we have
$f(x)=g(x) \times q(x)+r(x)$
$x^{5}-4 x^{3}+x^{2}+3 x+1=\left(x^{3}-3 x+1\right)\left(x^{2}-1\right)+2$
$x^{5}-4 x^{3}+x^{2}+3 x+1=x^{5}-3 x^{3}+x^{2}-x^{3}+3 x-1+2$
$x^{5}-4 x^{3}+x^{2}+3 x+1=x^{5}-3 x^{3}-x^{3}+x^{2}+3 x-1+2$
$x^{5}-4 x^{3}+x^{2}+3 x+1=x^{5}-4 x^{3}+x^{2}+3 x+1$
Hence an example for polynomial $f(x), g(x), q(x)$ and $r(x)$ satisfying $f(x)=g(x) \times q(x)+r(x)$ are
$f(x)=x^{5}-4 x^{3}+x^{2}+3 x+1$
$g(x)=\left(x^{3}-3 x+1\right)$
$q(x)=\left(x^{2}-1\right)$
$r(x)=2$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.