using pascal’s triangle, write out the expansion of (a+1)^3

Answer 1

To expand $(a+1{)}^3$ using Pascal's Triangle, we first identify the coefficients from the third row of Pascal's Triangle. The rows of Pascal's Triangle start with row 0 at the top:

• Row 0: $1$
• Row 1: $11$
• Row 2: $121$
• Row 3: $1331$

The coefficients for the expansion of $(a+b{)}^n$ are found in row $n$. Since we are expanding $(a+1{)}^3$, we use row 3: $1$, $3$, $3$, $1$.

The expansion is given by:

$$(a+1{)}^3=1\cdot a^3+3\cdot a^2\cdot 1+3\cdot a\cdot 1^2+1\cdot 1^3$$

This simplifies to:

$$=a^3+3a^2+3a+1$$

So, the expansion of $(a+1{)}^3$ is:

$$a^3+3a^2+3a+1$$