Question

Use the pascals triangle to expand
(5x+y)^5

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Answer 1

Answer: 3125x^5 + 3125x^4y + 1250x^3y^2 + 250x^2y^3 + 25xy^4 + y^5

Explanation:

Apply the Binomial Theorem

for exponential 5

we use 1 5 10 10 5 1

raise each power in descending for a and increasing for b as follow (ax)^5(b)y^0 + 5(ax)^4 (y)^1 ... (ax)^0 (by)^5

NOTE THAT you may not always have an a or b, it can also be a fraction or negative yet use the same method.

(5x)^5 (y)^0 + 5(5x)^4 (y)^1 + 10(5x)^3 (y)^2 + 10(5x)^2 (y)^3 + 5(5x)^1 (y)^4 + (5x)^0 (y)^5

Thus answer is

3125x^5 + 3125x^4y + 1250x^3y^2 + 250x^2y^3 + 25xy^4 + y^5

Answer 2

Explanation:

Apply the Binomial Theorem:

We know that
`(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5`

So substituting a=5x and b=y into our equation will give the desired result.
Our new expansion is
`5^5x^5+5*5^4x^4y + 10*5^3x^3y^2+10*5^2x^2y^3+5*5xy^4+y^5`

Simplifying our coefficients yields
`3125x^5+3125x^4y+1250x^3y^2+250x^2y^3+25xy^4+y^5`