Maths - Binomial Theorem
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\[(a + b)^12\]What would you use to expand this?? The binomial theorem.
\[(a + b)^12\]
What would you use to expand this?
The binomial theorem.
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\[1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1\]What is the next row of Pascal’s triangle??
\[1 \quad 6 \quad 15 \quad 20 \quad 15 \quad 6 \quad 1\]What is the $r$-th term in the binomial expansion of $(a + b)^n$?
\[1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1\]
What is the next row of Pascal’s triangle?
What does the $n$-th row of Pascal’s triangle start with (ignoring the top)?
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\[\left(\begin{matrix} 8 \\ 3 \end{matrix}\right)\]Because of the symmetry property, what is this equal to??
\[\left(\begin{matrix} 8 \\\\ 5 \end{matrix}\right)\]#####
\[\left(\begin{matrix} n \\ r \end{matrix}\right)\]Because of the symmetry property, what is this equal to??
\[\left(\begin{matrix} n \\\\ n-r \end{matrix}\right)\]What are the coefficients for $(a + b)^3$?
2021-10-14
What is $(a + b)^3$?
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.
\[\left(\begin{matrix} 8 \\ 3 \end{matrix}\right)\]
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?
When is the expansion for
\[(1 + x)^n\]valid??
\[|x| < 1\]When is the expansion for
\[(1 + 4x)^n\]valid??
\[|x| < \frac{1}{4}\]When is the approximation for
\[(1 + x)^n\]the best?? When the values of $x$ are small.
2021-10-22
What is
\[(a + bx)^n\]equivalent too??
\[a^n\left( 1 + \frac{b}{a}x \right)^n\]How would you tackle finding the binomial expansion for
\[\frac{4 - 5x}{(1 + x)(2 - x)}\]?? Use partial fractions.
2022-01-11
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.