Multiply the given polynomials.

(i) $2 x; x^{2} - 2 x - 1$ (ii) $x^{5} - 1; x^{3} + 2 x^{2} + 2$ (iii) $2 y + 1; y^{2} - 2 y^{3} + 3 y$

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Solution

(i)
$2 x (x^{2} - 2 x - 1) = 2 x \times x^{2} + 2 x \times (- 2 x) + 2 x \times (- 1) = 2 x^{3} - 4 x^{2} - 2 x$
(ii)
$(x^{5} - 1) \times (x^{3} + 2 x^{2} + 2) = x^{5} (x^{3} + 2 x^{2} + 2) - 1 (x^{3} + 2 x^{2} + 2) = x^{8} + 2 x^{7} + 2 x^{5} - x^{3} - 2 x^{2} - 2$
(iii)
$(2 y + 1) \times (y^{2} - 2 y^{3} + 3 y) = 2 y (y^{2} - 2 y^{3} + 3 y) + 1 (y^{2} - 2 y^{3} + 3 y) = 2 y^{3} - 4 y^{4} + 6 y^{2} + y^{2} - 2 y^{3} + 3 y = - 4 y^{4} + 2 y^{3} - 2 y^{3} + 6 y^{2} + y^{2} + 3 y = - 4 y^{4} + 7 y^{2} + 3 y$

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Factorise :
(i) $x^{3}$ −2 $x^{2}$ −x+2
(ii) 2 $y^{3}$ + $y^{2}$ −2y−1

(x^{3} - 2 x^{2} + 3 x - 4) (x - 1) - (2 x - 3) (x^{2} - x + 1)

x^{2} (x - y) y^{2} (x + 2 y)

\frac{x}{3} + \frac{y}{4} = 4; \frac{x}{2} - \frac{y}{4} = 1

\frac{x}{3} + 5 y = 13; 2 x + \frac{y}{2} = 19

\frac{2}{x} + \frac{3}{y} = 13; \frac{5}{x} - \frac{4}{y} = - 2

Divide the given polynomial by the given monomial:
$(2 y^{3} + 3 y^{2} + 12 y) \div 3 y$

Factorise:

(i) x³ − 2x² − x + 2 (ii) x³ + 3x² −9x − 5

(iii) x³ + 13x² + 32x + 20 (iv) 2y³ + y² − 2y − 1

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