How do I use Pascal's triangle to expand ${(x + 2)}^{5}$ $(x + 2)^5$ ?

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How do I use Pascal's triangle to expand ${(x + 2)}^{5}$ $(x + 2)^5$ ?

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Answer 1 · 2015-07-09T22:18:59

You write out the sixth 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, 5, 10, 10, 5, 1.

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

${(x + y)}^{5} = x^{5} + 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} y^{3} + 5 x y^{4} + y^{5}$ $(x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5$

But our polynomial is ${(x + 2)}^{5}$ $(x+2)^5$ .

${(x + 2)}^{5} = x^{5} + 5 x^{4} \times 2 + 10 x^{3} \times 2^{2} + 10 x^{2} \times 2^{3} + 5 x \times 2^{4} + 2^{5}$ $(x+2)^5 = x^5 + 5x^4×2 + 10x^3×2^2 + 10x^2×2^3 + 5x×2^4 + 2^5$

${(x + 2)}^{5} = x^{5} + 10 x^{4} + 40 x^{3} + 80 x^{2} + 80 x + 32$ $(x+2)^5 = x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32$

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How do I use Pascal's triangle to expand ${(x + 2)}^{5}$ $(x + 2)^5$ ?

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