How do I use Pascal's triangle to expand ${(2 x + y)}^{4}$ $(2x + y)^4$ ?

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How do I use Pascal's triangle to expand ${(2 x + y)}^{4}$ $(2x + y)^4$ ?

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Answer 1 · 2015-07-09T21:59:37

Write out the fifth row of Pascal's triangle and make the appropriate substitutions.

Explanation:

Pascal's triangle is

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The numbers in the fifth row are 1, 4, 6, 4, 1.

They are the coefficients of the terms in a fourth order polynomial.

${(x + y)}^{4} = x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}$ $(x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

But our polynomial is ${(2 x + y)}^{4}$ $(2x+y)^4$ .

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

${(2 x + y)}^{4} = 16 x^{4} + 32 x^{3} y + 24 x^{2} y^{2} + 8 x y^{3} + y^{4}$ $(2x+y)^4 = 16x^4 + 32x^3y + 24x^2y^2 + 8xy^3 + y^4$

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How do I use Pascal's triangle to expand ${(2 x + y)}^{4}$ $(2x + y)^4$ ?

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