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How do you find the coefficient of $x$ in the expansion of ${(x + 3)^5}$?

Last updated date: 26th Feb 2024
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IVSAT 2024
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Hint: We are going to solve this question by first mentioning the formula that we will be using in this question. Then mention all the given terms by comparing values from the question. And then finally evaluate the coefficient of $x$.

Complete step-by-step solution:
First we will start off by using the binomial theorem which is given by \[{(x + y)^n} = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}}
  n \\
\end{array}} \right){x^{n - k}}{y^k}} \] where \[\left( {\begin{array}{*{20}{c}}
  n \\
\end{array}} \right) = \dfrac{{n!}}{{k!(n - k)!}}\]
Now, here for $n = 5$ and $y = 3$ we are looking for the coefficient of ${x^1}$. This means we need
  n - k = 1 \\
  \,\,\,\,\,\,\,k = 4 \\
Now, if we substitute the values in the above mentioned formula, then we get,
   = \left( {\begin{array}{*{20}{c}}
  5 \\
\end{array}} \right){x^1}{.3^4} \\
   = \dfrac{{5!}}{{4!1!}}.81.x \\
   = 405x \\
Hence, the coefficient of $x$ in the expansion of ${(x + 3)^5}$ is \[405x\].

Additional information: The binomial theorem or expansion describes the algebraic expansion of a binomial. According to the theorem, it is possible to expand the polynomial ${(x + y)^n}$ into a sum involving terms of the form $a{x^b}{y^c}$, where the exponents $b$ and $c$ are non negative integers with $b + c = n$, and the coefficient of $a$ of each term is a specific positive integer depending on $n$ and $b$. The coefficients that appear in the binomial expansion are called binomial coefficients. The binomial coefficient can be interpreted as the number of ways to choose $k$ elements from an n-element set. As the power increases the expansion becomes lengthy and tedious to calculate. A binomial expression that has been raised to a very large power can be easily calculated with the help of Binomial theorem.

Note: While relating such terms, make sure you relate along with the powers and the respective signs. While substituting any terms, substitute such that the expression becomes easier to solve. While mentioning any formula, always check if there are any exceptions.