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## Q. 10.5.2

Using the Binomial Theorem

Expand:        $(x+2)^4$.

## Verified Solution

We use the Binomial Theorem

$(a+b)^n=\left(\begin{array}{l}n \\0\end{array}\right) a^n+\left(\begin{array}{l}n \\1\end{array}\right) a^{n-1} b+\left(\begin{array}{l}n \\2\end{array}\right) a^{n-2} b^2+\left(\begin{array}{l}n \\3\end{array}\right) a^{n-3} b^3+\cdots+\left(\begin{array}{l}n \\n\end{array}\right) b^n$

to expand $(x+2)^4$. In $(x+2)^4$, a = x, b = 2, and n = 4. In the expansion, powers of x are in descending order, starting with $x^4$. Powers of 2 are in ascending order, starting with $2^0$. (Because $2^0 = 1$, a 2 is not shown in the first term.) The sum of the exponents on x and 2 in each term is equal to 4 , the exponent in the expression $(x+2)^4$.  $=x^4+8 x^3+24 x^2+32 x+16$