Question

I have five questions from Polynomial Identities and the Binomial Theorem. Could someone be of service. Medal and fan!!(:

Answer 1

It says to expand the binomial (2x^2+y^2)^4.

Answer 2

Then it tells me to fill in the missing coefficients 16x^8+( )x^6y^2+( )x^4y^4+8x^2y^6+y8

Answer 3

I found the two answers for that I think the first is 32 and the second is 24.

Answer 4

The binomial theorem says
\[(a+b)^n=\sum_{k=0}^n\binom nk a^{n-k}b^k\]
Set \(n=4\), \(a=2x^2\) and \(b=y^2\). This gives
\[(2x^2+y^2)^4=\sum_{k=0}^4\binom 4k (2x^2)^{4-k}(y^2)^k\]
The first term (when \(k=0\)) is
\[\color{red}{\binom 40} \color{blue}{(2x^2)^{4-0}}\color{green}{(y^2)^0}=\color{red}1\times\color{blue}{2^4x^8}\times\color{green}1=16x^8\]
The second term (\(k=1\)) is
\[\color{red}{\binom 41}\color{blue}{(2x^2)^{4-1}}\color{green}{(y^2)^1}=\color{red}4\times\color{blue}{2^3x^6}\times\color{green}{y^2}=\underbrace{?}x^6y^2\]

Answer 5

Right, so the first coefficient you needed to find is 32. The next (24) is also right. There's a distinct pattern to the coefficients.
\[Cx^{\alpha}y^{\beta}~~\implies~~C=\binom 4k2^{4-k}\]

Answer 6

What does it mean if there is an exclamation point within an expanded form of a binomial.

Answer 7

The exclamation point is used to denote the factorial function:
\[n!=\begin{cases}n\times(n-1)\times(n-2)\cdots2\times1&\text{for }n\ge1\\
1&\text{for }n=0\\
0&\text{for }n<0\end{cases}\]

Answer 8

So if I want to find \(5!\), I would compute
\[5!=5\times4\times3\times2\times1=120\]

Answer 9

would it make sense if you had this?