Class : 10 : Algebra

Algebra (बीजगणित)

(H.C.F, L.C.M, Radical & Surd, Indices, Algebraic fraction, Equation)

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$\underline{\textbf{Important formulae:}}$

$ 1. \space (a+b)^2=a^2+2ab+b^2\\ 2. \space (a-b)^2=a^2-2ab+b^2\\ 3. \space (a+b)^3=a^3+3a^2b+3ab^2+b^3\\ 4. \space (a-b)^3=a^3-3a^2b+3ab^2-b^3\\ 5. \space a^2-b^2=(a+b)(a-b)\\ 6. \space a^3+b^3=(a+b)(a^2-ab+b^2)\\ 7. \space a^3-b^3=(a-b)(a^2+ab+b^2)\\ 8. \space a^4+a^2b^2+b^4=(a^2+ab+b^2)(a^2-ab+b^2)\\ 9. \space a^2+b^2=(a+b)^2 - 2ab \space Or \space (a-b)^2+2ab $

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1. Highest Common Factor [H.C.F](म.स.):

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2. Lowest Common Multiple [L.C.M](ल.स.)

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$\large \textbf{Example:}$ ल.स. (L.C.M) पत्ता लगाउनुहाेस् ।

a. $x^3-1, x^3-x^2$ र $x^4+x^2+1$

Here,

प.अ. = $x^3-1$
= $x^3-1^3$
= $(x-1)(x^2+x+1)$
दाे.अ. = $x^3-x^2$
= $x^2(x-1)$
ते.अ. = $x^4+x^2+1$
= $x^4+1+x^2$
= $(x^2)^2+1^2+x^2$
= $(x^2)^2+2.x^2.1+1^2-2.x^2.1+x^2$
= $(x^2+1)^2-2x^2+x^2$
= $(x^2+1)^2-x^2$
= $(x^2+1-x)(x^2+1+x)$
= $(x^2-x+1)(x^2+x+1)$
साझा गुणन खण्ड = $(x-1)(x^2+x+1)$
बाँकी गुणन खण्ड = $x^2(x^2+1-x)$
ल.स. = साझा गुणन खण्ड $\times$ बाँकी गुणन खण्ड
= $(x-1)(x^2+x+1) \times x^2(x^2+1-x)$
= $x^2(x-1)(x^2+1-x)(x^2+x+1)$ ans.

b. $(\frac{x^2}{y^2})^2+1+(\frac{y^2}{x^2})^2$ र $(\frac{x}{y})^3+(\frac{y}{x})^3$

Here,

प.अ. = $(\frac{x^2}{y^2})^2 + 1 + (\frac{y^2}{x^2})^2$ = $(\frac{x^2}{y^2})^2 + (\frac{y^2}{x^2})^2 + 1$
= $(\frac{x^2}{y^2})^2 + 2. \frac{x^2}{y^2} . \frac{y^2}{x^2} + (\frac{y^2}{x^2})^2 - 2. \frac{x^2}{y^2} . \frac{y^2}{x^2} + 1$
= $(\frac{x^2}{y^2} + \frac{y^2}{x^2})^2 - 2 + 1$
= $(\frac{x^2}{y^2} + \frac{y^2}{x^2})^2 - 1^2$
=$(\frac{x^2}{y^2} + \frac{y^2}{x^2} + 1)(\frac{x^2}{y^2} + \frac{y^2}{x^2} - 1)$
=$(\frac{x^2}{y^2} + 1 + \frac{y^2}{x^2})(\frac{x^2}{y^2} - 1 + \frac{y^2}{x^2})$
दो.अ. = $(\frac{x}{y})^3 + (\frac{y}{x})^3$ = $(\frac{x}{y} + \frac{y}{x})(\frac{x^2}{y^2} - \frac{x}{y}.\frac{y}{x} + \frac{y^2}{x^2})$
= $(\frac{x}{y} + \frac{y}{x})(\frac{x^2}{y^2} - 1 + \frac{y^2}{x^2})$
साझा गुणन खण्ड = $(\frac{x^2}{y^2} - 1 + \frac{y^2}{x^2})$
बाँकी गुणन खण्ड = $(\frac{x}{y} + \frac{y}{x})(\frac{x^2}{y^2} + 1 + \frac{y^2}{x^2})$
ल.स. = $(\frac{x^2}{y^2} - 1 + \frac{y^2}{x^2})(\frac{x}{y} + \frac{y}{x})(\frac{x^2}{y^2} + 1 + \frac{y^2}{x^2})$
= $(\frac{x}{y} + \frac{y}{x})(\frac{x^2}{y^2} - 1 + \frac{y^2}{x^2})(\frac{x^2}{y^2} + 1 + \frac{y^2}{x^2})$ ans.

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3. Radical and Surd (साधारण मूलक र सर्ड)

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4. Indices (घाताङ्क)

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5. Algebraic Fraction (बीजीय भिन्न)

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6. Equation (समीकरण)

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