How do you expand ${(a + 2 b)}^{4}$ $(a+2b)^4$ ?

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How do you expand ${(a + 2 b)}^{4}$ $(a+2b)^4$ ?

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Answer 1 · 2017-07-16T18:09:39

$a^{4} + 8 a^{3} b + 24 a^{2} b^{2} + 32 a b^{3} + 16 b^{4}$ $a^4+8a^3b+24a^2b^2+32ab^3+16b^4$

Explanation:

$we can expand using the appropriate row of coefficients$ $"we can expand using the appropriate row of coefficients"$
$from Pascal's triangle$ $"from "color(blue)"Pascal's triangle"$

$for n = 4 the coefficients are$ $"for " n=4" the coefficients are"$

$\times \times \times 1 x 4 x 6 x 4 x 1$ $color(white)(xxxxxx)1color(white)(x)4color(white)(x)6color(white)(x)4color(white)(x)1$

$with decreasing powers of a from a^{4} \to a^{0}$ $"with decreasing powers of "a" from "a^4toa^0$

$and increasing powers of 2 b from {(2 b)}^{0} \to {(2 b)}^{4}$ $"and increasing powers of "2b" from "(2b)^0to(2b)^4$

$\Rightarrow {(a + 2 b)}^{4}$ $rArr(a+2b)^4$

$= 1 . a^{4} {(2 b)}^{0} + 4 . a^{3} {(2 b)}^{1} + 6 . a^{2} {(2 b)}^{2} + 4 . a {(2 b)}^{3}$ $=1.a^4(2b)^0+4.a^3(2b)^1+6.a^2(2b)^2+4.a(2b)^3$
$= + 1 . {(2 b)}^{4}$ $color(white)(=)+1.(2b)^4$

$= a^{4} + 8 a^{3} b + 24 a^{2} b^{2} + 32 a b^{3} + 16 b^{4}$ $=a^4+8a^3b+24a^2b^2+32ab^3+16b^4$

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How do you expand ${(a + 2 b)}^{4}$ $(a+2b)^4$ ?

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