Question

Which is the correct expansion of (r 3s)³ using the binomial theorem expression?

1) (r³ + 3r²s + 3rs² + s³)
2) (r⁹ + 3r⁶s³ + 3r³s⁶ + s⁹)
3) (r⁶ + 3r⁴s² + 3r²s⁴ + s⁶)
4) (r²⁷ + 3r¹8s⁹ + 3r⁹s¹8 + s²⁷)

Answer 1

The correct expansion of (r + 3s)^3 using the binomial theorem expression is r^3 + 3r^2s + 3rs^2 + s^3. This adheres to the binomial coefficients and powers for each term in the expansion.

Step-by-step explanation:

When expanding the expression (r + 3s)^3 using the binomial theorem, each term in the expansion is found using the combination of the exponents that add up to the power of the binomial, which in this case is 3. The binomial expansion formula (a + b)^n = a^n + n(a^(n-1))b + n(n-1)/2(a^(n-2))b^2 + ... + b^n can be used for this purpose.

The correct expansion is the first option:

1. r^3 + 3r^2s + 3rs^2 + s^3

This is because when you expand (r + 3s)^3, you will have one r term raised to the third power, three terms where r is squared and s is to the first power, three terms where r is to the first power and s is squared, and finally one term where s is to the third power. The coefficients correspond to the binomial coefficients, which for the cubic case are 1, 3, 3, and 1.

Therefore, the expanded form using the binomial theorem is r^3 + 3r^2s + 3rs^2 + s^3.