Question 33 Find the following products-i x-2-2 yx2-4-x y-4 y2ii x 2-1 x 4- x 2-1

(i) ( x/2+2y ) ( x^2/4 xy+4y^2 )=x/2 ( x^2/4 xy+4y^2 )+2y ( x^2/4 xy+4y^2 ) x/2 ( x^2/4 )+x/2( xy)+x/2(4y^2)+2y ( x^2/4 ) +2y( xy)+2y(4y^2) x^3/8 x^2y/2+2x ...

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Standard IX

Mathematics

Algebraic Identities

Question 33 F...

Question

Question 33
Find the following products:
(i) $(\frac{x}{2} + 2 y) (\frac{x^{2}}{4} - x y + 4 y^{2})$

(ii) $(x^{2} - 1) (x^{4} + x^{2} + 1)$

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Solution

(i) $(\frac{x}{2} + 2 y) (\frac{x^{2}}{4} - x y + 4 y^{2}) = \frac{x}{2} (\frac{x^{2}}{4} - x y + 4 y^{2}) + 2 y (\frac{x^{2}}{4} - x y + 4 y^{2})$
$\frac{x}{2} (\frac{x^{2}}{4}) + \frac{x}{2} (- x y) + \frac{x}{2} (4 y^{2}) + 2 y (\frac{x^{2}}{4}) + 2 y (- x y) + 2 y (4 y^{2})$
$\frac{x^{3}}{8} - \frac{x^{2} y}{2} + 2 x y^{2} + \frac{x^{2} y}{2} - 2 x y^{2} + 8 y^{3}$
$= \frac{x^{3}}{8} + 8 y^{3}$

(ii) $(x^{2} - 1) (x^{4} + x^{2} + 1) = x^{2} (x^{4} + x^{2} + 1) - 1 (x^{4} + x^{2} + 1)$
$= [x^{2} (x^{4}) + x^{2} (x^{2}) + x^{2} (1) - 1 (x)^{4} - 1 (x^{2}) - 1 (1)]$
$= x^{6} + x^{4} + x^{2} - x^{4} - x^{2} - 1 = x^{6} - 1$

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Question 33
Find the following products:
(i) $(\frac{x}{2} + 2 y) (\frac{x^{2}}{4} - x y + 4 y^{2})$

(ii) $(x^{2} - 1) (x^{4} + x^{2} + 1)$

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(i) (3x + 2y) (9x² − 6xy + 4y²)

(ii) (4x − 5y) (16x² + 20xy + 25y²)

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

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

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

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

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

(ix) (1 − x) (1+ x + x²)

(x) (1 + x) (1 − x + x²)

(xi) (x² − 1) (x⁴ + x² + 1)

(xii) (x³ + 1) (x⁶ − x^{3 +}1)

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

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(\frac{x}{2} - \frac{2}{5}) (\frac{2}{5} - \frac{x}{2}) - x^{2} + 2 x

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

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

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

Q. Find the following product:

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

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Question 33 Find the following products: (i) (x2+2y)(x24−xy+4y2) (ii) (x2−1)(x4+x2+1)

(i) (x2+2y)(x24−xy+4y2)=x2(x24−xy+4y2)+2y(x24−xy+4y2) x2(x24)+x2(−xy)+x2(4y2)+2y(x24)+2y(−xy)+2y(4y2) x38−x2y2+2xy2+x2y2−2xy2+8y3 =x38+8y3 (ii) (x2−1)(x4+x2+1)=x2(x4+x2+1)−1(x4+x2+1) =[x2(x4)+x2(x2)+x2(1)−1(x)4−1(x2)−1(1)] =x6+x4+x2−x4−x2−1=x6−1

Question 33
Find the following products:
(i) $(\frac{x}{2} + 2 y) (\frac{x^{2}}{4} - x y + 4 y^{2})$

(ii) $(x^{2} - 1) (x^{4} + x^{2} + 1)$