To expand (x + 3)^7 using the binomial theorem, we can think of the theorem as a way to express the expansion of a binomial raised to a power in a systematic manner. The fundamental concept behind the binomial theorem is that it allows us to expand expressions of the form (a + b)^n, where ‘a’ and ‘b’ are any numbers or variables and ‘n’ is a non-negative integer.
In our case, we are dealing with (x + 3)^7. First, we identify ‘a’ as ‘x’ and ‘b’ as ‘3’, and our exponent ‘n’ is ‘7’. The binomial theorem states that the expansion will consist of terms formed by the coefficients from Pascal’s triangle, multiplied by the appropriate powers of ‘a’ and ‘b’. Each term in the expansion can be described as:
- The coefficient, which is derived from the chosen row of Pascal’s triangle corresponding to the exponent ‘n’.
- The first variable (‘a’, which is ‘x’) raised to a power decreasing from ‘n’ to ‘0’.
- The second variable (‘b’, which is ‘3’) raised to a power increasing from ‘0’ to ‘n’.
As we expand the expression, we will have a total of ‘n + 1’ terms. For (x + 3)^7, this means we’ll have 8 terms in our final expansion. The coefficients for each term can be calculated or looked up in Pascal’s triangle based on the exponent ‘7’.
In summary, to expand (x + 3)^7 using the binomial theorem, we utilize the concept of coefficients from Pascal’s triangle, combine powers of ‘x’ and ‘3’, and systematically write out the terms to form our complete expansion.