Question

Verify division algorithm i.e., Dividend = Divisor $\times$ Quotient + Remainder, in each of the following. Also, write the quotient and remainder :

$\begin{array}{ccc}\left(i\right)& 14x^2+13x-15& 7x-4\\ \left(ii\right)& 15z^2-20z^2+13z-12& 3z-6\\ \left(iii\right)& 6y^5-28y^3+3y^2+30y-9& 2y^2-6\\ \left(iv\right)& 34x-22x^3-12x^4-10x^2-75& 3x+7\\ \left(v\right)& 15y^4-16y^3+9y^2-\frac{10}{3}y+6& 3y-2\\ \left(vi\right)& 4y^3+8y+8y^2+7& 2y^2-y+1\\ \left(vii\right)& 6y^5+4y^4+4y^3+7y^2+27y+6& 2y^3+1\end{array}$

Answer 1

(i) Dividend = $14x^2+13x-15$ and Divisor $=7x-4$ Now, dividing the dividend by divisor, we get

Quotient $=2x+3$

and Remainder = - 3

(ii) Dividend $=15z^3-20z^2+13z-12$ and divisor = 3z - 6

Now dividing the dividend by divisor, we get

Quotient $=5z^2+\frac{10}{3}z+11$

and Remainder = 54

(iii) Dividend $=6y^5-28y^3+3y^2+30y-9$

divisor $=2y^2-6$

Now, dividing the dividend by divisor, we get

Quotient $=3y^3-5y+\frac{3}{2}$

Remainder = 0

(iv) Dividend $=34x-22x^3-12x^4-10x^2-75$

$=-12x^4-22x^3-10x^2+34x-75$

and divisor $=3x+7$

Now, dividing the dividend by divisor, we get

Quotient $=-4x^3+2x^2-8x+30$

Remainder = -285

(v) Dividend $=15y^4-16y^3+9y^2-\frac{10}{3}y+6$ and divisor = 3y - 2

Now, dividing the dividend by divisor, we get

Quotient $=5y^3-2y^2+\frac{5}{3}y$

Remainder = 6

(vi) Dividend $=4y^3+8y+8y^2+7$

$=4y^3+8y^2+8y+7$

and divisor $=2y^2-y+1$

Now, dividing the dividend by divisor, we get

Quotient $2y+5$

Remainder $=11y+2$

(vii) Dividend $=6y^5+4y^4+4y^3+7y^2+27y+6$

Divisor $=2y^3+1$

Now dividing the dividend by the divisor, we get

Quotient $=3y^2+2y+2$

Remainder $=4y^2+25y+4$