Question

When a polynomial $p\left(x\right)$ of degree $3$ is divided by $3x^2-8x+5$ quotient and reminder obtained are linear polynomials such that $p\left(1\right)=19$ and $p\left(\frac{5}{3}\right)=25$. So, find the reminder polynomial.

Answer 1

Let $p\left(x\right)=(3x^2-8x+5)(ax+b)+(cx+d)$

where, $(ax+b)$ is quotient & $(cx+d)$ is remainder

$p\left(1\right)=19$

$\therefore \left(3\right(1{)}^2-8\left(1\right)+5\left)\right(a\left(1\right)+b)+(c\left(1\right)+d)=19$

$\therefore c+d=19$ ………..$\left(1\right)$

$P\left(\frac{5}{3}\right)=25$

$\therefore (3{\left(\frac{5}{3}\right)}^2-8\left(\frac{5}{3}\right)+5)(a\left(\frac{5}{3}\right)+b)+(c\left(\frac{5}{3}\right)+d)=25$

$\therefore \frac{5c}{3}+d=25$

$\therefore 5c+3d=75$ ……….$\left(2\right)$

$\therefore$ From $\left(1\right)$ & $\left(2\right)$,

$\therefore C=9$ and $d=10$

$\therefore$ remainder is $9x+10$.