verify if the relationship is true? (a+b)^5-a^5-b^5=5ab(a+b)(a^2+ab+b^2)

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to verify if (a+b)^5 - a^5 - b^5 = 5ab(a+b)(a^2+ab+b^2)

We can expand (a + b)^5 = a^5 + 5a^4b + 10 a^3b^2 + 10 a^2b^3 + 5ab^4 + b^5

The left hand side

(a+b)^5 - a^5 - b^5

=> a^5 + 5a^4b + 10 a^3b^2 + 10 a^2b^3 + 5ab^4 + b^5 - a^5 - b^5

=> 5a^4b + 10 a^3b^2 + 10 a^2b^3 + 5ab^4

=> 5ab(a^3 + 2a^2b + 2ab^2 + b^3)

=> 5ab(a + b)( a^2 + ab + b^2)

which is the right hand side

We prove that (a+b)^5 - a^5 - b^5 = 5ab(a+b)(a^2+ab+b^2).

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll expand the binomial:

(a+b)^5=a^5+5a^4*b+10a^3*b^2+10a^2*b^3+5b^4*a+b^5

We'll subtract a^5 and b^5 from expansion and we'll get:

5a^4*b+10a^3*b^2+10a^2*b^3+5b^4*a

We'll combine the middle terms and the extremes and we'll factorize them:

5ab(a^3 + b^3) + 10a^2*b^2(a+b)

But the sum of cubes is:

a^3  +b^3 = (a+b)(a^2 - ab + b^2)

5ab(a+b)(a^2 - ab + b^2) + 10a^2*b^2(a+b)

We'll factorize by 5ab(a+b)

5ab(a+b)(a^2 - ab + b^2 + 2ab)

We'll combine like terms inside brackets:

5ab(a+b)(a^2 + ab + b^2)=5ab(a+b)(a^2 + ab + b^2) q.e.d.

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