How do I use Pascal's triangle to expand a binomial?

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How do I use Pascal's triangle to expand a binomial?

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Answer 1 · 2015-10-31T09:31:53

Rows of Pascal's triangle provide the coefficients to expand ${(a + b)}^{n}$ $(a+b)^n$ as follows...

Explanation:

To expand ${(a + b)}^{n}$ $(a+b)^n$ look at the row of Pascal's triangle that begins $1, n$ $1, n$ . This provides the coefficients.

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For example, ${(a + b)}^{4} = a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}$ $(a+b)^4 = a^4+4a^3b+6a^2b^2+4ab^3+b^4$ from the row $1, 4, 6, 4, 1$ $1, 4, 6, 4, 1$

How about ${(2 x - 5)}^{4}$ $(2x-5)^4$ ?

Let $a = 2 x$ $a = 2x$ and $b = - 5$ $b = -5$ .

Then:

${(2 x - 5)}^{4} = {(a + b)}^{4} = a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}$ $(2x-5)^4 = (a+b)^4 = a^4+4a^3b+6a^2b^2+4ab^3+b^4$

$= {(2 x)}^{4} + 4 {(2 x)}^{3} (- 5) + 6 {(2 x)}^{2} {(- 5)}^{2} + 4 (2 x) {(- 5)}^{3} + {(- 5)}^{4}$ $=(2x)^4+4(2x)^3(-5)+6(2x)^2(-5)^2+4(2x)(-5)^3+(-5)^4$

$= 16 x^{4} + 4 (8 x^{3}) (- 5) + 6 (4 x^{2}) (25) + 4 (2 x) (- 125) + (625)$ $=16x^4+4(8x^3)(-5)+6(4x^2)(25)+4(2x)(-125)+(625)$

$= 16 x^{4} - 160 x^{3} + 600 x^{2} - 1000 x + 625$ $=16x^4-160x^3+600x^2-1000x+625$

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How do I use Pascal's triangle to expand a binomial?

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