Question

How do you use pascal’s triangle to expand ${\left( {x - 2} \right)^4}$

Answer 1

Hint: Given a binomial expression and we have to use the pascal triangle to expand the expression. In this method, we have to start the pascal triangle by writing the number $1$ in the first row. Then write number $1$ in the next row two times. Then each new row is obtained by writing number $1$ at the start and end and the remaining numbers of each row are calculated by writing the sum of two numbers which lies left and right in the above row of that position. In the expansion of expression ${\left( {a + b} \right)^n}$, the power of $a$ decreases from $n$ to zero as we move from left to right whereas the power of $b$ increases from zero to $n$. Now select the coefficients of the triangle starting from $1$, $n$.

Formula used:
The pascal’s triangle is
$
  {\text{ }}1 \\
  {\text{ }}1{\text{ }}1 \\
  {\text{ }}1{\text{ }}2{\text{ }}1 \\
  {\text{ }}1{\text{ }}3{\text{ }}3{\text{ }}1 \\
  {\text{ }}1{\text{ }}4{\text{ }}6{\text{ }}4{\text{ }}1 \\
  1{\text{ }}5{\text{ }}10{\text{ }}10{\text{ }}5{\text{ }}1 \\
$
Complete step-by-step answer:
Step 1: We are given the binomial expression ${\left( {x - 2} \right)^4}$. Here the value of exponent is equal to $4$ which means we will pick the coefficients from the fourth row of the pascal’s triangle.
$\left( {1,4,6,4,1} \right)$
Next as we move from left to right terms in the pascal’s triangle then the power of $x$ decreases from $4$ to zero and the power of $ - 2$ increases from zero to $4$.
$ \Rightarrow {\left( {x - 2} \right)^4} = 1{\left( x \right)^4}{\left( { - 2} \right)^0} + 4{\left( x \right)^3}{\left( { - 2} \right)^1} + 6{\left( x \right)^2}{\left( { - 2} \right)^2} + 4{\left( x \right)^1}{\left( { - 2} \right)^3} + 1{\left( x \right)^0}{\left( { - 2} \right)^4}$
Step 2: Now we will write the expression by multiplying the values in the expression.
$ \Rightarrow {\left( {x - 2} \right)^4} = {x^4} - 8{x^3} + 24{x^2} - 32x + 16$

Hence the expansion of the expression is ${x^4} - 8{x^3} + 24{x^2} - 32x + 16$

Note:
In such types of questions students mainly make mistakes while picking the coefficients of the relevant row of the pascal’s triangle. In these types of questions, the first row will be counted as row zero, then row one, and so on, and pick the row number equivalent to the exponent of the binomial exponent.