Question

Expand (x – 3)^5 using the Binomial Theorem and Pascal’s triangle. Show all necessary steps.

Answer 1

The expansion of  (x – 3)^5 is     $x^5-15x^4+90x^3-270x^2+405x-243$

The Binomial Theorem is the method of expanding an expression that has been raised to any finite power. A binomial Theorem is a powerful tool of expansion, which has application in Algebra, probability, etc. Binomial Expression: A binomial expression is an algebraic expression that contains two dissimilar terms.

Pascal's Triangle is a never-ending equilateral triangle in which the arrays of numbers arranged in a triangular manner. The triangle starts at 1 and continues placing the number below it in a triangular pattern

Using Binomial theorem,

=$\sum _{i=0}^5\binom{5}{i}x^{\left(5-i\right)}\left(-3\right)^i$

=$x^5-15x^4+90x^3-270x^2+405x-243$

Using Pascal Triangle  for (x-1)

                                               1

                                       1    1              

           1    2    1            

        1    3    3    1        

     1    4    6    4    1      

  1    5    10    10    5    1  

1    6    15    20    15    6    1

Accordingly replacing 1 with 3 we get

= $x^5-15x^4+90x^3-270x^2+405x-243$

Thus the expansion of  (x – 3)^5 is      $x^5-15x^4+90x^3-270x^2+405x-243$

Learn more about Binomial Theorem here :

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