To expand the given binomial, we should recall the Binomial Theorem. It states that for every positive integer n, we can expand the expression (a+b)^n by using the numbers in the n^(th) row of Pascal's Triangle.
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(a+b)^n= & P_0a^nb^0+P_1a^(n-1)b^1
& +
& ...
& +
& P_(n-1)a^1b^(n-1)+P_na^0b^n
In the above formula, P_0, P_1, ..., P_n are the numbers in the n^(th) row of Pascal's Triangle.
Row 1.3cm Pascal's Triangle 1.2cm
cccccccccccc
0 & & & & & & 1 & & & & &
1 & & & & & 1 & & 1 & & & &
2 & & & & 1 & & 2 & & 1 & & &
3 & & & 1 & & 3 & & 3 & & 1 & &
4 & & 1 & & 4 & & 6 & & 4 & & 1 &
5 & 1 & & 5 & & 10 & & 10 & & 5 & & 1
Note that each number greater than 1 found in the triangle is the sum of the two numbers diagonally above it. Now consider the given binomial.
( 4+ 9x)^3
We can substitute the first term for a and the second term for b using the Binomial Theorem equation and the coefficients from Pascal's Triangle.
