Expand Using the Binomial Theorem (3x+y)^5

Problem

(3*x+y)^5

Solution

1. Identify the components of the binomial (a+b)^n where a=3*x b=y and n=5

2. Apply the Binomial Theorem formula, which states (a+b)^n=(∑_k=0^n)((n/k))*a^(n−k)*b^k)

3. Write out the terms using the combinations (5/k)) for k=0,1,2,3,4,5

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

4. Calculate the binomial coefficients using Pascal's triangle or the formula (n/k)=(n!)/(k!(n−k)!)) which are $1, 5, 10, 10, 5, 1$.

5. Evaluate the powers of 3*x for each term.

1*(243*x^5)*(1)+5*(81*x^4)*(y)+10*(27*x^3)*(y^2)+10*(9*x^2)*(y^3)+5*(3*x)*(y^4)+1*(1)*(y^5)

6. Simplify each term by multiplying the coefficients.

243*x^5+405*x^4*y+270*x^3*y^2+90*x^2*y^3+15*x*y^4+y^5

Final Answer

(3*x+y)^5=243*x^5+405*x^4*y+270*x^3*y^2+90*x^2*y^3+15*x*y^4+y^5

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