Question

How do you use the Binomial Theorem to expand `(5x + 2y)^6`?

Answer 1

`15625x^6 + 37500x^5y + 37500x^4y^2 + 20000x^3y^3 + 6000x^2y^4 +960xy^5 + 64y^6`

Explanation:

The Binomial Theorem states that
`(a + b)^n = ""_nC_0a^nb^0 + ""_(n-1)C_1a^(n-1)b^1 + ""_(n-2)C_2a^(n-2)b^2 +`..... `""_0C_n a^0b^n`
where `""_(n-x)C_x =(n!)/((n-x)!x!)`

`:. (5x+2y)^6 =`
`(6!)/(6!0!)(5x)^6 +(6!)/(5!1!)(5x)^5(2y)+(6!)/(4!2!) (5x)^4(2y)^2 +(6!)/(3!3!)(5x)^3(2y)^3 + (6!)/(2!4!)(5x)^2(2y)^4 +(6!)/(1!5!)(5x)(2y)^5 +(6!)/(0!6!)(2y)^6`

`=1*5^6x^6 +6(5^5)x^5*2y + 15*5^4x^4*2^2y^2 + 20*5^3x^3*2^3y^3 + 15*5^2x^2*2^4y^4 + 6*5x*2^5y^5 +1*2^6y^6`

`=15625x^6 + 37500x^5y + 37500x^4y^2 + 20000x^3y^3 + 6000x^2y^4 +960xy^5 + 64y^6`