Question

How do you expand ${{\left( 2x+y \right)}^{4}}$ using Pascal’s triangle?

Answer 1

Here we have to expand ${{\left( 2x+y \right)}^{4}}$ by using Pascal’s triangle. The rows of Pascal’s triangle provide the coefficient to expand ${{\left( a+b \right)}^{2}}$ we consider here ${{\left( a+b \right)}^{n}}$ then to expand ${{\left( a+b \right)}^{n}}$ look at the row of Pascal’s triangle that begin $1$ in this provides the coefficient of terms. So by applying this we have to solve this problem.

Complete step-by-step answer:
First of we have to write out the ${{5}^{th}}$ row of Pascal’s triangle as a sequence:
Then, the sequence is written as,
$1,4,6,4,1...(i)$
After that write out descending powers of $2$ from ${{2}^{4}}$ to ${{2}^{0}}$ as a sequence.
Then, the sequence is written as,
$16,8,4,2,1...(ii)$
Now, we have two sequences.
Multiply the sequence $(i)$ and $(ii)$ we get,
The sequence is,
$16,32,24,8,1$
These are the coefficient of the expression:
${{\left( 2x+y \right)}^{4}}=16{{x}^{4}}+32{{x}^{3}}y+24{{x}^{2}}{{y}^{2}}+8x{{y}^{3}}+{{y}^{4}}$
Hence,
After combining the ${{5}^{th}}$ row of Pascal’s triangle and a sequence of power of $2$ for finding the expansion. Therefore the expansion is

${{\left( 2x+y \right)}^{4}}=16{{x}^{4}}+32{{x}^{3}}y+24{{x}^{2}}{{y}^{2}}+8x{{y}^{3}}+{{y}^{4}}$

Additional Information:
The binomial theorem is a kind of formula which helps us to expand binomials raised to the power of any number using Pascal's triangle. This is also known as a binomial theorem. In other ways the binomial theorem used Pascal's triangle for determining the coefficient, which describes the algebraic expansion of powers of a binomial. It shows what happens when you multiply a binomial by itself. For example let’s take a expression ${{\left( 4x+4 \right)}^{7}}$ It takes so much time to multiply the binomial seven times as it has power of $7.$
But the formula of the binomial theorem gives us an easy way on a shortcut that gives the expanded form of this expression. According to the given theorem, it is possible to expand power ${{\left( x+y \right)}^{n}}$ into a sum in the form $a{{x}^{6}}{{y}^{c}}.$
Where as the exponent $b$ and $c$ are positive integer with $b+c=n,$ and the coefficient $0,$ of each term is a positive integer deepens on $n$ and $b.$ If exponent is zero then the given power is committed from the term.

Note:
Use the binomial formula and Pascal’s triangle to expand a binomial which is raised to a power and also find its coefficient. The property for binomial expression having the number of terms is the one more than (exponent) and the sum of exponent in each term adds up to $n.$ The power first should start with $n$ and decreases by $1$ each term and second starts with $0$. It increases by $1$ each term. For example ${{\left( a+b \right)}^{4}}$ then $a$ is first tern and $b$ is second tern.