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Coefficients of (x+3y)^4

User Ktheory
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Answer:

To expand the expression (x + 3y)^4, we can use the binomial theorem. The general formula for expanding (a + b)^n is:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n

In this case, a = x and b = 3y, and n = 4. Let's expand the expression:

(x + 3y)^4 = C(4, 0) * x^4 * (3y)^0 + C(4, 1) * x^3 * (3y)^1 + C(4, 2) * x^2 * (3y)^2 + C(4, 3) * x^1 * (3y)^3 + C(4, 4) * x^0 * (3y)^4

Now, let's calculate the coefficients for each term:

C(4, 0) = 1

C(4, 1) = 4

C(4, 2) = 6

C(4, 3) = 4

C(4, 4) = 1

Substituting these coefficients back into the expanded expression:

(x + 3y)^4 = 1 * x^4 * 3^0 + 4 * x^3 * 3^1 + 6 * x^2 * 3^2 + 4 * x^1 * 3^3 + 1 * x^0 * 3^4

Simplifying further:

(x + 3y)^4 = x^4 + 12x^3y + 54x^2y^2 + 108xy^3 + 81y^4

Therefore, the expanded form of (x + 3y)^4 is x^4 + 12x^3y + 54x^2y^2 + 108xy^3 + 81y^4.

User Patrick Burns
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