Question

Find the following products: 

(i) (3x + 2y)(9x² – 6xy + 4y²) 

(ii) (4x – 5y)(16x² + 20xy + 25y²) 

(iii) (7p⁴ + q)(49p⁸ – 7p⁴q + q²) 

(iv) \((\frac{x}{2} + 2y)\) \((\frac{x^2}{4} - xy + 4y^2)\)

(v) \((\frac{3}{x} - \frac{5}{y}) (\frac{9}{x^2} + \frac{25}{y^2} + \frac{15}{xy})\)

(vi) \((3 + \frac{5}{x})\) \((9 - \frac{15}{x} + \frac{25}{x^2})\)

(vii) \((\frac{2}{x} + 3x)\) \((\frac{4}{x^2} + 9x^2 - 6)\)

(viii) \((\frac{3}{x} - 2x^2)\) \((\frac{9}{x^2} + 4x^4 - 6x)\)

(ix) (1 – x)(1 + x + x²) 

(x) (1 + x)(1 – x + x²) 

(xi) (x² – 1)(x⁴ + x² +1) 

(xii) (x³ + 1)(x⁶ – x³ + 1)

Answer 1

(i) (3x + 2y)(9x² – 6xy + 4y²)

= (3x + 2y)[(3x)² – (3x)(2y) + (2y)²)] [a³ + b³ = (a + b)(a² + b² – ab)]

= (3x)³ + (2y)³ 

= 27x³ + 8y³ 

(ii) (4x – 5y)(16x² + 20xy + 25y²) 

= (4x – 5y)[(4x)² + (4x)(5y) + (5y)²)]  [a³ – b³ = (a – b)(a² + b² + ab)]

= (4x)³ – (5y)³ 

= 64x³ – 125y³ 

(iii) (7p⁴ + q)(49p⁸ – 7p4q + q²) 

= (7p⁴ + q)[(7p⁴)² – (7p⁴)(q) + (q)²)] [a³ + b³ = (a + b)(a² + b² – ab)] 

= (7p⁴)³ + (q)³ 

= 343p¹² + q³ 

(iv)  \((\frac{x}{2} + 2y)\) \((\frac{x^2}{4} - xy + 4y^2)\)

[a³ – b³ = (a – b)(a² + b² + ab)] 

 \((\frac{x}{2} + 2y)\) \((\frac{x^2}{4} - xy + 4y^2)\)

(v)  \((\frac{3}{x} - \frac{5}{y}) (\frac{9}{x^2} + \frac{25}{y^2} + \frac{15}{xy})\)

(vi)  \((3 + \frac{5}{x})\) \((9 - \frac{15}{x} + \frac{25}{x^2})\)

(vii)  \((\frac{2}{x} + 3x)\) \((\frac{4}{x^2} + 9x^2 - 6)\)

(viii)  \((\frac{3}{x} - 2x^2)\) \((\frac{9}{x^2} + 4x^4 - 6x)\)

(ix) (1 – x)(1 + x + x²) 

[a³ – b³ = (a – b)(a² + b² + ab)] 

(1 – x)(1 + x + x²) can be written as,

(1 – x)[(1² + (1)(x)+ x²)] 

= (1)³ – (x)³ 

= 1 – x³ 

(x) (1 + x)(1 – x + x²) 

[a³ + b³ = (a + b)(a² + b² – ab)] 

(1 + x)(1 – x + x²) can be written as, 

(1 + x)[(1² – (1)(x) + x²)] 

= (1)³ + (x)³ 

= 1 + x³ 

(xi) (x² – 1)(x⁴ + x² +1) 

(x² – 1)[(x²)² – 12 + (x²)(1)] 

= (x²)³ – 13 

= x⁶ – 1 [a³ – b³ = (a – b)(a² + b² + ab)] 

(xii) (x³ + 1)(x⁶ – x³ + 1)

(x³ + 1)[(x³)² – (x³)(1) + 1²] 

= (x³)³ + 1³ 

= x⁹ + 1