Question

Apply division algorithm to find the quotient and remainder on dividing the polynomial 3 x^ + 4 x^ square + 6 x + 9 by x^ + 3x + 7 also verify the division algorithm

Answer 1

I suppose you mean

$\dfrac{3x^3+4x^2+6x+9}{x^2+3x+7}$

$3x^3=3x\cdot x^2$, and if we multiply $x^2+3x+7$ by $3x$ we get $3x^3+9x^2+21x$. Subtracting this from the numerator gives a remainder of $-5x^2-15x+9$.

$-5x^2=-5\cdot x^2$, and multiplying $x^2+3x+7$ by $-5$ gives $-5x^2-15x-35$. Subtracting this from the previous remainder gives a new remainder of $44$.

$44$ has no remaining factors of $x^2$ in it, so we're done, and

$\dfrac{3x^3+4x^2+6x+9}{x^2+3x+7}=\underbrace{3x-5}_{\rm quotient}+\dfrac{\overbrace{44}^{\rm remainder}}{x^2+3x+7}$