Question

What is the third term in the binomial expansion of (3x+y^3)^4

Answer 1

Answer:

The last choice is the one you want.

Step-by-step explanation:

Use the 5th row of Pascal's Triangle.  Since you have a 4th degree polynomial, there will be 5 terms in it.  The 5 coefficients, in order, are:

1, 4, 6, 4, 1

We will use these coefficients only up to and including the third one, since that is the one you want.  Binomial expansion using Pascal's Triangle looks like this:

$1(3x)^4(y^3)^0+4(3x)^3(y^3)^1+6(3x)^2(y^3)^2+...$

That third term is the one we are interested in.  That simplification gives us:

$6(9x^2)(y^6)$

Multiply 6 and 9 to get 54, and a final term of:

$54x^2y^6$

Answer 2

The third term of the given binomial expansion is $54(x^{2})(y^{5})$

What is binomial expansion?

The binomial expansion is based on a theorem that specifies the expansion of any power $(a+b)^{m}$ of a binomial (a + b) as a certain sum of products $a^{i} b^{i}$, such as (a + b)² = a² + 2ab + b².

How to find the third term in the binomial expansion of  (3x+y^3)^4 ?

• We know that the binomial expansion of  $(a+b)^{m}$ can be written as $mC_{0}(a^{m-0}) +mC_{1}(a^{m-1})b+ mC_{2}(a^{m-2})b^{2}+..................+mC_{m}b^{m}$
• So the (r+1)th term will be $mC_{r}(a^{m-r})b^{r}$

The given term is $(3x + y^{3}) ^{4}$

The third term in the expansion will be

$4C_{2}(9x^{2})(y^{3})^{2}\\ = 54(x^{2})(y^{5})$

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