Maths - Binomial Theorem
Pearson Edexcel Mathematics 2022
The binomial theorem and general term
\[(a + b)^{12}\]
What would you use to expand this?
The binomial theorem.
What is the $r$-th term in the binomial expansion of $(a + b)^n$?
Pascal’s triangle
\[1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1\]
What is the next row of Pascal’s triangle?
What does the $n$-th row of Pascal’s triangle start with (ignoring the top)?
Symmetry of binomial coefficients
\[ \left(\begin{matrix} 8 \\ 3 \end{matrix}\right) \]
Because of the symmetry property, what is this equal to?
\[ \left(\begin{matrix} n \\ r \end{matrix}\right) \]
Because of the symmetry property, what is this equal to?
Finding a particular term
What would be the expression for working out the $x^3$ term of $(2x + 6)^7$?
The binomial series for (1 + x)^n
What is the formula for $(1 + x)^n$ where $ \vert x \vert < 1$?
Why does the result
\[(1 + x)^n \equiv 1 + nx + \frac{n(n-1)}{2!}x^2 + ...\]
hold when $n > 1$ even though the sequence is infinite?
Because you get $0$ in the numerator for later terms and so they disappear.
What is the $x^2$ term in the formula for $(1 + x)^n$?
What is the $x^7$ term in the formula for $(1 + x)^n$?
Validity of the binomial series
When is the expansion for
\[(1 + x)^n\]
valid?
When is the expansion for
\[(1 + 4x)^n\]
valid?
When is the approximation for
\[(1 + x)^n\]
the best?
When the values of $x$ are small.
Manipulating into standard form
What is
\[(a + bx)^n\]
equivalent to?
How would you tackle finding the binomial expansion for
\[\frac{4 - 5x}{(1 + x)(2 - x)}\]
?
Use partial fractions.
Expanding a quotient under a root
How would you tackle finding the binomial expansion for
\[\sqrt{\frac{1-x}{1+4x}}\]
?
Treat it as $(1-x)^{\frac{1}{2}}(1+4x)^{-\frac{1}{2}}$ and multiply the expansions together.