Binomial Theorem

The One Thing

For a positive integer $n$, the binomial theorem expands $(a+b{)}^n$ into $n+1$ organised terms. The general term $$T_{r+1}=\left(\genfrac{}{}{0ex}{}{n}{r}\right)a^{n-r}{b}^r$$ allows particular terms, coefficients, middle terms, and terms independent of a variable to be found without expanding the entire expression.

Definitions and Results

• Binomial: An algebraic expression containing two terms connected by addition or subtraction, such as $a+b$ or $x-2$.

• Binomial Theorem: For a positive integer $n$,

$$(a+b{)}^n=\sum _{r=0}^n(\genfrac{}{}{0ex}{}{n}{r})a^{n-r}{b}^r.$$ Equivalently, $$(a+b{)}^n=(\genfrac{}{}{0ex}{}{n}{0})a^n+(\genfrac{}{}{0ex}{}{n}{1})a^{n-1}b+(\genfrac{}{}{0ex}{}{n}{2})a^{n-2}{b}^2+\cdots +(\genfrac{}{}{0ex}{}{n}{n})b^n.$$

• General Term: The term at position $r+1$ in the expansion is

$$T_{r+1}=\left(\genfrac{}{}{0ex}{}{n}{r}\right)a^{n-r}{b}^r,$$ where $r=0,1,2,\dots ,n$.

• Binomial Coefficient: The number $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$, also written $C(n,r)$, is

$$\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}.$$

• Factorial: For a non-negative integer $n$,

$$n!=n(n-1)(n-2)\cdots 3\cdot 2\cdot 1,$$ with $0!=1$.

• Pascal’s Triangle: A triangular arrangement in which each interior number is the sum of the two numbers directly above it. Its rows give the binomial coefficients.

• Number of Terms: The expansion of $(a+b{)}^n$ contains $n+1$ terms.

• Powers: The powers of $a$ decrease from $n$ to $0$, while the powers of $b$ increase from $0$ to $n$. The sum of the powers of $a$ and $b$ in every term is $n$.

• Symmetry of Coefficients:

$$\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\left(\genfrac{}{}{0ex}{}{n}{n-r}\right).$$ Important special cases are $$\left(\genfrac{}{}{0ex}{}{n}{0}\right)=\left(\genfrac{}{}{0ex}{}{n}{n}\right)=1,\left(\genfrac{}{}{0ex}{}{n}{1}\right)=\left(\genfrac{}{}{0ex}{}{n}{n-1}\right)=n.$$

• Pascal Identity:

$$\left(\genfrac{}{}{0ex}{}{n}{r}\right)+\left(\genfrac{}{}{0ex}{}{n}{r+1}\right)=\left(\genfrac{}{}{0ex}{}{n+1}{r+1}\right).$$

• Middle Term: If $n$ is even, there is one middle term,

$$T_{\frac{n}{2}+1}.$$ If $n$ is odd, there are two middle terms, $$T_{\frac{n+1}{2}}\text{and}{T}_{\frac{n+3}{2}}.$$

• Term Independent of a Variable: A term that does not contain a specified variable. It is found by setting the exponent of that variable in the general term equal to zero.

• Expansion of $(a-b{)}^n$: Replace $b$ by $-b$:

$$(a-b{)}^n=\sum _{r=0}^n(\genfrac{}{}{0ex}{}{n}{r})a^{n-r}(-b{)}^r.$$ Consequently, the signs alternate.

• First Expansions:

$$(a+b{)}^2=a^2+2ab+b^2,$$ $$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3.$$

• Sum of Coefficients: The sum of all coefficients in $(a+b{)}^n$ is obtained by putting $a=1$ and $b=1$:

$$(1+1{)}^n=2^n.$$

• Alternating Sum of Coefficients: For $n>0$, putting $a=1$ and $b=-1$ gives

$$(1-1{)}^n=0.$$

• Coefficient of a Particular Term: The coefficient of $a^{n-r}{b}^r$ is $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$. If variables or numerical factors occur inside the binomial, the general term must first be simplified before the coefficient is identified.

Worked Methods

1. Expanding a Binomial Completely

• Identify $a$, $b$, and the positive integer index $n$.
• Use

$$(a+b{)}^n=\sum _{r=0}^n(\genfrac{}{}{0ex}{}{n}{r})a^{n-r}{b}^r.$$

• Write the terms in order, decreasing the power of $a$ and increasing the power of $b$.
• Calculate the coefficients using factorials or Pascal’s triangle.
• Check that there are $n+1$ terms and that the powers in every term add to $n$.

For example, $$(a+b{)}^2=a^2+2ab+b^2$$ and $$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3.$$

2. Finding a Particular Term

• Determine the required position of the term.
• Relate the position to $r$: the term at position $r+1$ is $T_{r+1}$.
• Substitute $r$ into

$$T_{r+1}=\left(\genfrac{}{}{0ex}{}{n}{r}\right)a^{n-r}{b}^r.$$

• Simplify the coefficient and the powers.

The position is always one more than the value of $r$.

3. Finding a Particular Coefficient

• Write the general term

$$T_{r+1}=\left(\genfrac{}{}{0ex}{}{n}{r}\right)a^{n-r}{b}^r.$$

• Compare the required power with the corresponding exponent in the general term.
• Solve for $r$.
• Substitute the resulting value into $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$.
• If numerical factors or variables occur inside the binomial, simplify the complete term before identifying the coefficient.

The coefficient of $a^{n-r}{b}^r$ is $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$.

4. Finding a Middle Term

• Determine whether $n$ is even or odd.
• If $n$ is even, use

$$T_{\frac{n}{2}+1}.$$

• If $n$ is odd, use the two terms

$$T_{\frac{n+1}{2}}\text{and}{T}_{\frac{n+3}{2}}.$$

• Evaluate the required term or terms using the general term.

5. Finding a Term Independent of a Variable

• Write the general term.
• Simplify it sufficiently to identify the exponent of the specified variable.
• Set that exponent equal to zero.
• Solve for $r$.
• Substitute the resulting $r$ into the general term and simplify.

A term independent of a variable is therefore found by making that variable’s exponent zero.

6. Expanding a Difference of Two Terms

• Rewrite the expression as $(a+(-b){)}^n$.
• Apply the binomial theorem:

$$(a-b{)}^n=\sum _{r=0}^n(\genfrac{}{}{0ex}{}{n}{r})a^{n-r}(-b{)}^r.$$

• Since $(-b{)}^r$ is positive for even $r$ and negative for odd $r$, retain the alternating signs.

7. Using Pascal’s Triangle

• Identify the required row, remembering that the expansion of $(a+b{)}^n$ uses the row corresponding to $n$.
• Read the coefficients from left to right.
• Match them with powers of $a$ decreasing from $n$ to $0$ and powers of $b$ increasing from $0$ to $n$.
• Use the symmetry of the row when appropriate:

$$\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\left(\genfrac{}{}{0ex}{}{n}{n-r}\right).$$

8. Finding the Sum or Alternating Sum of Coefficients

For the sum of coefficients:

• Set $a=1$ and $b=1$.
• Evaluate

$$(1+1{)}^n=2^n.$$

For the alternating sum:

• Set $a=1$ and $b=-1$.
• For $n>0$, evaluate

$$(1-1{)}^n=0.$$

Where It Goes Wrong

• Confusing the position of a term with $r$: $T_{r+1}$ is the term at position $r+1$, not position $r$.
• Reversing the powers: the power of $a$ decreases from $n$ to $0$, while the power of $b$ increases from $0$ to $n$.
• Forgetting that every term must have total degree $n$, so the powers of $a$ and $b$ must add to $n$.
• Using the wrong coefficient by overlooking the required value of $r$; the coefficient of $a^{n-r}{b}^r$ is $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$.
• Treating $(a-b{)}^n$ as though all signs remain positive instead of replacing $b$ by $-b$, which produces alternating signs.
• Forgetting the conditions for middle terms: even $n$ gives one middle term, whereas odd $n$ gives two.
• Failing to set the specified variable’s exponent equal to zero when finding a term independent of that variable.
• Applying the theorem outside the stated setting: the material concerns positive integral indices $n$.

What Gets Asked

• Expand $(a+b{)}^n$ using the binomial theorem.
• Write the general term $T_{r+1}$ of an expansion.
• Find a specified term by its position.
• Find the coefficient of a specified power of $a$, $b$, or another variable.
• Determine the middle term or middle terms.
• Find the term independent of a specified variable.
• Expand expressions of the form $(a-b{)}^n$ and account for alternating signs.
• Calculate binomial coefficients using factorials or Pascal’s triangle.
• Use the symmetry

$$\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\left(\genfrac{}{}{0ex}{}{n}{n-r}\right)$$ or Pascal’s identity $$\left(\genfrac{}{}{0ex}{}{n}{r}\right)+\left(\genfrac{}{}{0ex}{}{n}{r+1}\right)=\left(\genfrac{}{}{0ex}{}{n+1}{r+1}\right).$$

• Find the sum of all coefficients, using $2^n$.
• Find the alternating sum of coefficients, using $0$ for $n>0$.
• Explain why the theorem has $n+1$ terms and why each term has total degree $n$.