Question

Divide the first polynomial by the second. Explain the steps.

(1) 21b; 7
(2) 36p²; 6p
(3) 5a³; a³
(4) −30n²; 3n
(5) 15m³; 5m
(6) 2b⁵; b²
(7) 28x⁴; −7x²
(8) −8y⁵; y³

Answer 1

(1) 21b; 7 i.e. 21b ÷ 7
Dividend = 21b = 7 × 3 × b
Divisor = 7
$\frac{21b}{b}=\frac{7\times 3\times b}{7}$
= 3×b
∴ 21b ÷ 7 = 3b

(2) 36p²; 6p i.e. 36p² ÷ 6p
Dividend = 36p²
= 6 × 6 × p × p
Divisor = 6 p = 6 × p
$\frac{36p^2}{p}=\frac{6\times 6\times p\times p}{6\times p}$
= 6×p
∴ 36p² ÷ 6p = 6p

(3) 5a³; a³ i.e. 5a³ ÷ a³
Dividend = 5a³ = 5 × a × a× a
Divisor = a³ = a × a × a
$\frac{5a^3}{a^3}=\frac{5\times a\times a\times a}{a\times a\times a}$
= 5
∴ 5a³ ÷ a³ = 5

(4) −30n²; 3n i.e. −30n²÷ 3n
Dividend = −30n² = −5 × 3 × 2 × n× n
Divisor = 3n = 3 × n
$\frac{-30n^2}{3n}=\frac{-5\times 3\times 2\times n\times n}{3\times n}$
= −10
∴ −30n² ÷ 3n = −10 n

(5) 15m³; 5m i.e. 15m³ ÷ 5 m
Dividend = 15m³ = 5 × 3 × m × m× m
Divisor = 5m = 5 × m
$\frac{15m^3}{5m}=\frac{5\times 3\times m\times m\times m}{5\times m}$
= 3m²
∴ 15m³ ÷ 5 m = 3m²

(6) 2b⁵; b² i.e. 2b⁵ ÷ b²
Dividend = 2b⁵ = 2 × b × b × b× b × b
Divisor = b² = b × b
$\frac{2b^5}{b^2}=\frac{2\times b\times b\times b\times b\times b}{b\times b}$
= 2b³
∴ 2b⁵ ÷ b² = 2b³

(7) 28x⁴; −7x² i.e. 28x⁴ ÷ −7x²
Dividend = 28x⁴ = 2 × 2 × 7 × x × x × x × x
Divisor = −7x² = −7× x × x
$\frac{28x^4}{-7x^2}=\frac{2\times 2\times 7\times x\times x\times x\times x}{-7\times x\times x}$
= −4x²
∴ 28 x⁴÷ −7x² = −4x²

(8) −8y⁵; −y³ i.e. −8y⁵ ÷ −y³
Dividend = −8y⁴ = − 2 × 2 × 2 × 2 × y × y × y × y
Divisor = −y² = − y × y
$\frac{-8y^5}{-y^3}=\frac{-2\times 2\times 2\times y\times y\times y\times y\times y}{-y\times y\times y}$
= 8y²
∴ −8y⁵ ÷ −y² = 8y²