Find the last 3 terms in the expansion of (3𝑥 + 𝑥2)5 . Simplify your answer.

Answer 1

If your problem is to determine the last 3 terms in the expansion of (3x + x²)⁵ then a good way to do that is to use Pascal's Triangle:

You can look that up but suffice it to say that the row in Pascal's Triangle for the 5th power has coefficients of 1, 5, 10, 10, 5,

These coefficients combined with a contribution of the first term (3x) and the last term (x²) get applied to each term. The contribution of the first term is, for term 1 of the expansion (1st term)⁵. And for the second term and on down, we reduce the power by 1. The contribution of the last term is, for term 6 of the expansion (Last term)⁵. And for the next term to the left, we reduce the power by 1. This table shows the examples of this problem:

Term

Number          1               2               3               4               5               6

Pascal’s

Triangle          1               5               10             10             5               1

Coeff

1st Term         (3x)⁵         (3x)⁴         (3x)³         (3x)²         (3x)¹         (3x)⁰

Contribution   243x⁵        81x⁴          27x³          9x²            3x             1

Last Term       (x²)⁰          (x²)¹          (x²)²          (x²)³          (x²)⁴          (x²)⁵

Contribution   1               x²              x⁴              x⁶              x⁸              x¹⁰

Term               243x⁵        405x⁶        270x⁷        90x⁸          15x⁹         x¹⁰

To find the term value, you multiply the 3 parts together (Pascal's Triangle coefficient, 1st term contribution, and last term contribution)