Question

What is the third term in the binomial expansion of (3x + y3)4?

54x^2*y^3
18x^2*y^3
18x^2*y^6
54x^2*y^6

Answer 1

The third term in the binomial expansion of (3x + y^3)^4 is 18x^2 * y^6, which is obtained by applying the binomial theorem to the binomial expression.

Step-by-step explanation:

To find the third term in the binomial expansion of (3x + y^3)^4, we use the binomial theorem, which provides a formula for expanding expressions raised to a power. The general binomial expansion is given by (a + b)^n = a^n + (n choose 1)a^(n-1)b + (n choose 2)a^(n-2)b^2 + ... + (n choose n)b^n. In this case, we want the third term, which corresponds to the k=2 term of the expansion (since the first term corresponds to k=0).

Following the formula for the binomial coefficient (n choose k) = n! / (k!(n-k)!) and binomial expansion, the third term is given by:

(4 choose 2) * (3x)^(4-2) * (y^3)^2 = (4*3/2*1) * 9x^2 * y^6 = 18x^2 * y^6.

Therefore, the third term in the binomial expansion of (3x + y^3)^4 is 18x^2 * y^6.

Answer 2

`18x^2y^6` that is option third is correct

Step-by-step explanation:

We have been given with the expression
`(3x+y^3)^4`

We have general formula for binomial expansion which is

`a^n +na^(n-1)b+ ((n)(n-1))/(2) a^(n-2)b^2+----------+b^n`

Here
`a= 3x , b=y^3and\\n=4`

Substituting the values in the formula we will get

`3x^4+4(3x)^(4-1)y^3+ ((4)(4-1))/(2) 3x^(4-2) (y^(3))^2\\\\ (3x)^4+4(3x)^3y^3+6(3x)^2y^6\\\\(3x)^4+12x^3y^3+18x^2y^6`+----

we can clearly see that third term of the expansion will be

`18x^2y^6`

Therefore, Option third is correct