Full solution

Q. lation:

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Divide

1

st. d by

1

st termol

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\frac{5x^{2}}{x}=5x

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Simplify:

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(i)

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6(2x+y-7xy)-3(5x-2y+5xy)

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(iii)

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x(x^{2}+2xy+y^{2})+4y(x^{2}+3xy+9y^{2})

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(ii)

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4(2x-3y+xy)-5

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Find the product:

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2\mid03

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(i)

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(\sqrt{x}+\sqrt{y})(x-\sqrt{xy}+$\newline$y)

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(ii)

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(x^{3}-xy+y^{3})(x^{3}+xy+y:}

Distribute and Simplify (i): Simplify the expression (i) by distributing the multiplication over addition and subtraction. $\newline$ $6(2x + y - 7xy) - 3(5x - 2y + 5xy)$ $\newline$ $= 12x + 6y - 42xy - 15x + 6y - 15xy$ $\newline$ $= (12x - 15x) + (6y + 6y) - (42xy + 15xy)$ $\newline$ $= -3x + 12y - 57xy$
Distribute and Simplify (ii): Simplify the expression (ii) by distributing the multiplication over addition and subtraction. $\newline$ $4(2x - 3y + xy) - 5$ $\newline$ = $8x - 12y + 4xy - 5$ $\newline$ = $8x - 12y + 4xy - 5$ (No further simplification is possible)
Distribute and Simplify (iii): Simplify the expression (iii) by distributing the multiplication over addition. $\newline$ $x(x^2 + 2xy + y^2) + 4y(x^2 + 3xy + 9y^2)$ $\newline$ = $x^3 + 2x^2y + xy^2 + 4yx^2 + 12xy^2 + 36y^3$ $\newline$ = $x^3 + (2x^2y + 4yx^2) + (xy^2 + 12xy^2) + 36y^3$ $\newline$ = $x^3 + 6x^2y + 13xy^2 + 36y^3$
Find Product (i): Find the product of the given polynomials (i). $\newline$ $(\sqrt{x} + \sqrt{y})(x - \sqrt{xy} + y)$ $\newline$ $= \sqrt{x} \cdot x + \sqrt{x} \cdot (-\sqrt{xy}) + \sqrt{x} \cdot y + \sqrt{y} \cdot x + \sqrt{y} \cdot (-\sqrt{xy}) + \sqrt{y} \cdot y$ $\newline$ $= x^{3/2} - x\sqrt{y} + \sqrt{xy} + x^{3/2} - \sqrt{xy} + y^{3/2}$ $\newline$ $= 2x^{3/2} - x\sqrt{y} + \sqrt{xy} - \sqrt{xy} + y^{3/2}$
Find Product (ii): Find the product of the given polynomials (ii). $\newline$ $(x^3 - xy + y^3)(x^3 + xy + y^3)$ $\newline$ = $x^3 \cdot x^3 + x^3 \cdot xy + x^3 \cdot y^3 - xy \cdot x^3 - xy \cdot xy - xy \cdot y^3 + y^3 \cdot x^3 + y^3 \cdot xy + y^3 \cdot y^3$ $\newline$ = $x^6 + x^4y + x^3y^3 - x^4y - x^2y^2 - xy^4 + x^3y^3 + xy^4 + y^6$ $\newline$ = $x^6 + (x^4y - x^4y) + (x^3y^3 + x^3y^3) - x^2y^2 + (xy^4 - xy^4) + y^6$ $\newline$ = $x^6 + 2x^3y^3 - x^2y^2 + y^6$