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Long Division Method to Divide Two Polynomials

Divide. Write...

Question

Divide. Write the quotient and the remainder.
(a) $(21 x^{4} - 14 x^{2} + 7 x) \div 7 x^{3}$
(b) $(6 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2}) \div 2 x^{2}$
(c) $(25 m^{4} - 15 m^{3} + 10 m + 8) \div 5 m^{3}$

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Solution

Let us find quotient and the remainder by long division method:
(i) $3 x 7 x^{3}) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 21 x^{4} - 14 x^{2} + 7 x 21 x^{4} (-)________________- 14 x^{2} + 7 x$
Therefore, quotient $= 3 x$ and remainder $R = - 14 x^{2} + 7 x$
(ii)
$3 x^{3} - 2 x^{2} + 4 x + 1 2 x^{2}) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 6 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} 6 x^{5} (-) -- 4 x^{4} + 8 x^{3} + 2 x^{2} - 4 x^{2} (-) +8 x^{3} + 2 x^{2} 8 x^{3} (-) -_2 x^{2} (-) - 2 x^{2}_0$
Therefore, $Q = 3 x^{3} - 2 x^{2} + 4 x + 1$ and remainder $= 0$ .
(iii) $\frac{}{}$
$5 m - 3 5 m^{3}) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 25 m^{4} - 15 m^{3} + 10 m + 8 25 m^{4} (-) -- 15 m^{3} + 10 m + 8 - 15 m^{3} (-) +__________________10 m + 8$
Therefore, quotient $= 5 m - 3$ and remainder $= 10 m + 8$

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Q. Divide. Write the quotient and the remainder.
(1) 21m² ÷ 7m
(2) 40a³ ÷ (−10a)
(3) (−48p⁴) ÷ (−9p²)
(4) 40m⁵ ÷ 30m³
(5) (5x³ − 3x²) ÷ x²
(6) (8p³ − 4p²) ÷ 2p²
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(8) (21x⁴ − 14x² + 7x) ÷ 7x³
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(10) (25m⁴ − 15m³ + 10m + 8) ÷ 5m³

Q. Divides and write the quotient and the remainder.

({6 x}^{5} - {4 x}^{4} + {8 x}^{3} + {2 x}^{2}) \div {2 x}^{2}

(6 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{3}) \div 2 x^{2}

(6 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2}) \div 2 x^{2}

Enter 1 if it is true else 0.

(25 m^{4} - 15 m^{3} + 10 m + 8) \div 5 m^{3}

= 5 m - 3 + \frac{2}{m^{2}} + \frac{8}{5 m^{3}}

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