Question

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

Dividend		Divisor
(i)	14x2 + 13x − 15	7x − 4
(ii)	15z3 − 20z2 + 13z − 12	3z − 6
(iii)	6y5 − 28y3 + 3y2 + 30y − 9	2y2 − 6
(iv)	34x − 22x3 − 12x4 − 10x2 − 75	3x + 7
(v)	15y4 − 16y3 + 9y2 − $\frac{10}{3}$y + 6	3y − 2
(vi)	4y3 + 8y + 8y2 + 7	2y2 − y + 1
(vii)	6y5 + 4y4 + 4y3 + 7y2 + 27y + 6	2y3 + 1

Answer 1

(i)

Quotient = 2x + 3
Remainder = $-$3
Divisor = 7x $-$ 4
Divisor $\times$ Quotient + Remainder = (7x $-$ 4) (2x + 3) $-$ ​3
= 14x² + 21x $-$ 8x $-$ 12 $-$ ​3
= 14x² + 13x $-$ 15
= Dividend
Thus,
Divisor $\times$ Quotient + Remainder = Dividend
Hence verified.

(ii)

$$\begin{gathered} \mathrm{Quotient}=5z^2+\frac{10}{3}z+11 \\ \mathrm{Remainder}=54 \\ \mathrm{Divisor}=3z-6 \\ \mathrm{Divisor}\times \mathrm{Quotient}+\mathrm{Remainder}=(3z-6)\left(5z^2+\frac{10}{3}z+11\right)+54 \\ =15z^3+10z^2+33z-30z^2-20z-66+54 \\ =15z^3-20z^2+13z-12 \\ =\mathrm{Dividend} \\ \text{Thus}, \\ \mathrm{Divisor}\times \mathrm{Quotient}+\mathrm{Remainder}=\mathrm{Dividend} \end{gathered}$$
Hence verified.

(iii)


Quotient = $3y^3-5y+\frac{3}{2}$
Remainder = 0
Divisor = 2y² $-$ 6
Divisor $\times$ Quotient + Remainder =
$$\begin{gathered} (2y^2-6)\left(3y^3-5y+\frac{3}{2}\right)+0 \\ =6y^5-10y^3+3y^2-18y^3+30y-9 \\ =6y^5-28y^3+3y^2+30y-9 \end{gathered}$$
= Dividend

Thus, Divisor $\times$ Quotient + Remainder = Dividend
Hence verified.

(iv)

Quotient = $-$ 4x³ + 2x² $-$ 8x + 30
Remainder = $-$ 285
Divisor = 3x + 7
Divisor $\times$ Quotient + Remainder = (3x + 7) ($-$ 4x³ + 2x² $-$ 8x + 30) $-$ 285
= $-$ 12x⁴ + 6x³ $-$ 24x² + 90x $-$ 28x³ + 14x² $-$ 56x + 210 $-$ ​285
= $-$ 12x ⁴ $-$ 22x³ $-$ 10x² + 34x $-$ 75
= Dividend
Thus,
Divisor $\times$ Quotient + Remainder = Dividend
Hence verified.

(v)


Quotient = $5y^3-2y^2+\frac{5}{3}y$
Remainder = 6
Divisor = 3y $-$ 2
Divisor $\times$ Quotient + Remainder = (3y $-$ 2) (5y³ $-$ 2y² + $\frac{5}{3}y$) + 6
= $15y^4-6y^3+5y^2-10y^3+4y^2-\frac{10}{3}y+6$
= $15y^4-16y^3+9y^2-\frac{10}{3}y+6$
= Dividend
Thus,
Divisor $\times$ Quotient + Remainder = Dividend
Hence verified.

(vi)

Quotient = 2y + 5
Remainder = 11y + 2
Divisor = 2y² $-$ y + 1
Divisor $\times$ Quotient + Remainder = (2y² $-$ y + 1) (2y + 5) + 11y + 2
= 4y³ +10y² $-$ 2y² $-$ 5y + 2y + 5 + 11y + 2
= 4y³ + 8y² + 8y + 7
= Dividend
Thus,
Divisor $\times$ Quotient + Remainder = Dividend
Hence verified.

(vii)




Quotient = 3y² + 2y + 2
Remainder = 4y² + 25y + 4
Divisor = 2y³ + 1
Divisor $\times$ Quotient + Remainder = (2y³ + 1) (3y² + 2y + 2) + 4y² + 25y + 4
= 6y⁵ + 4y⁴ + 4y³ + 3y² + 2y + 2 + 4y² + 25y + 4
= 6y⁵ + 4y⁴ + 4y³ + 7y² + 27y + 6
= Dividend
Thus,
Divisor $\times$ Quotient + Remainder = Dividend
Hence verified.