Binomial theorem

Pascal’s triangle organizes the coefficients of $(a+b)^n$. Row $n=0$ is $1$. Each new row begins and ends with 1, and every interior entry is the sum of the two entries above it.

The first rows are $n=0: 1$; $n=1: 1,1$; $n=2: 1,2,1$; $n=3: 1,3,3,1$; $n=4: 1,4,6,4,1$; $n=5: 1,5,10,10,5,1$; and $n=6: 1,6,15,20,15,6,1$.

Every row is symmetric because $\binom nr=\binom n{n-r}$. The sum of all coefficients in row $n$ is $2^n$, found by substituting $a=b=1$ into $(a+b)^n$.

For $n\ge1$, the alternating sum of a row is zero. This follows by substituting $a=1,b=-1$, giving $(1-1)^n=0$.

The symbol $\binom nr$, read “$n$ choose $r$,” counts the ways to choose $r$ objects from $n$ when order does not matter. In the binomial theorem it is the coefficient of $a^{n-r}b^r$.

For integers $0\le r\le n$, $\binom nr=\frac{n!}{r!(n-r)!}$. The factorial $n!$ means $n(n-1)(n-2)\cdots2\cdot1$, with $0!=1$.

A GDC usually provides an $nCr$ command. Enter $n$, choose $nCr$, then enter $r$. Even when using technology, show which coefficient is being calculated.

To expand a binomial, identify the complete first and second components, choose the coefficients for row $n$, decrease the first power, increase the second power, then simplify each term.

Keep a negative component in parentheses as $(-k)^r$, with $k>0$. This factor is negative for odd $r$ and positive for even $r$. The sign of the whole term also depends on its other factors; for $(a-kx)^n$ with $a,k>0$, the coefficients of successive powers of $x$ alternate.

Arrange the final polynomial in descending powers of the chosen variable unless the question requests another order.

The term containing $b^r$ is $T_{r+1}=\binom nr a^{n-r}b^r$. The term number is $r+1$ because the first term corresponds to $r=0$.

To find the $p$th term, set $r=p-1$. This off-by-one adjustment is one of the most important details in binomial questions.

The general term lets us find one required term without writing the full expansion. It also exposes the power of a variable, which can be matched to a requested exponent.

To find the coefficient of $x^k$, write the general term, determine its power of $x$, set that exponent equal to $k$, and solve for the integer index $r$. Then evaluate only that term.

The coefficient does not include the variable factor. After finding the matching term, remove $x^k$ and report its numerical or parameter coefficient.

If the equation for $r$ gives a non-integer or a value outside $0\le r\le n$, no such term appears.

A constant term has variable exponent zero. In an expansion involving positive and negative powers of $x$, write the general term and collect the total exponent of $x$.

Set that exponent equal to zero and solve for $r$. A constant exists only if the solution is an integer in $0\le r\le n$.

After finding $r$, substitute it into the entire general term. Do not stop after solving the exponent equation; the constant coefficient still has to be evaluated.

Negative and fractional components follow the same finite theorem. Keep a negative component in parentheses so its powers produce the correct sign, and keep fractional powers of coefficients exact.

For $(a-kx)^n$ with $a,k>0$ and nonnegative integer $n$, coefficients alternate in sign with $r$, because of $(-k)^r$. For $(a+\frac{x}{m})^n$, with $m\ne0$, the factor $(\frac{x}{m})^r$ contributes $m^{-r}$ to each coefficient.

A GDC can calculate $nCr$, factorials, or individual terms. A CAS expansion may be useful as a check where permitted, but an IB response should show the theorem or general-term method.

Write $P(x)=c_0+c_1x+\cdots+c_nx^n$, including zero coefficients for missing powers. Then $P(1)=c_0+c_1+\cdots+c_n$, while $P(-1)=c_0-c_1+c_2-\cdots+(-1)^n c_n$; signs follow exponent parity, not displayed-term order.

Attempt each question before opening its solution. The problems progress from Pascal coefficients and full expansions to targeted terms, constants, parameters, and coefficient identities.

For targeted questions, avoid expanding everything. Write the general term, match the required power, verify the index, then simplify only the necessary term.