Expand the following formulas
$(a + b)^{2}$ =?
$(a - b)^{2}$ =?
$(a^{2} + b^{2})$ =?
$(a + b)^{3}$ =?
$(a - b)^{3}$ =?
$a^{3} + b^{3}$ =?
$a^{3} - b^{3}$ =?

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Solution

${(a + b)}^{2} = a^{2} + b^{2} + 2 a b$
${(a - b)}^{2} = a^{2} + b^{2} - 2 a b$
$(a^{2} + b^{2}) = {(a + b)}^{2} - 2 a b$
$= {(a - b)}^{2} + 2 a b$
${(a + b)}^{3} = a^{3} + b^{3} + 3 a b (a + b)$
$= a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$
${(a - b)}^{3} = a^{3} - b^{3} - 3 a b (a - b)$
$= a^{3} - b^{3} - 3 a^{2} b + 3 a b^{2}$
$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$
$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$

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Q. Expand

a^{3} b^{2}, a^{2} b^{3}, b^{2} a^{3}, b^{3} a^{2} .

Are they all same?

Q. Which of the following equations are correct?

(i) (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)

(i i) (a + b)^{3} = a^{3} + b^{3} + 3 a b (a - b)

(i i i) (a - b)^{3} = a^{3} - b^{3} - 3 a b (a + b)

(i v) a^{2} - b^{2} = (a + b) (a - b)

Which of the following equation/s is/are correct?

$i) (a + b + c)^{2} = a^{2} + b^{2} + c^{2} - 2 (a b + b c + c a)$
$i i) (a + b)^{3} = a^{3} + b^{3} + 3 a b (a - b)$
$i i i) (a - b)^{3} = a^{3} - b^{3} - 3 a b (a + b)$
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Q. Simplify the following expressions.

\frac{a^{2} - b^{2}}{a - b} - \frac{a^{3} - b^{3}}{a^{2} - b^{2}}

Q. If

x = 1 + a + a^{2} + a^{3} + . . . .

\infty (| a | < 1)

and

y = 1 + b + b^{2} + b^{3} + . . .

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Expand the following formulas(a+b)2=?(a−b)2=?(a2+b2)=?(a+b)3=?(a−b)3=?a3+b3=?a3−b3=?

(a+b)2=a2+b2+2ab(a−b)2=a2+b2−2ab(a2+b2)=(a+b)2−2ab=(a−b)2+2ab(a+b)3=a3+b3+3ab(a+b)=a3+b3+3a2b+3ab2(a−b)3=a3−b3−3ab(a−b)=a3−b3−3a2b+3ab2a3+b3=(a+b)(a2−ab+b2)a3−b3=(a−b)(a2+ab+b2)

Expand the following formulas
$(a + b)^{2}$ =?
$(a - b)^{2}$ =?
$(a^{2} + b^{2})$ =?
$(a + b)^{3}$ =?
$(a - b)^{3}$ =?
$a^{3} + b^{3}$ =?
$a^{3} - b^{3}$ =?