Question

expand (x-4)^5 using the binomial theorem and pascal’s triangle show all necesssry steps

Answer 1

The binomial expansion of the $(x-4)^5$ is  $x^5+20x^4+160x^3+640x^2+1280x+1024$

According to the statement

we have given that a equation and we have to expand it by use of the binomial theorem.

So, For this purpose we know that the

A binomial theorem, statement that for any positive integer n, the nth power of the sum of two numbers a and b may be expressed as the sum of n + 1 terms of the form.

And the general formula is

$(x+y)^n= ^nC_0x^ny^0+^nC_1x^(n-1)y^1+^nC_2x^(n-2)y^2+...+^nC_nx^0y^n$

And we have to solve it with the given equation (x-4)^5.

Then

$(x-4)^5=x^5+20x^4+(20)/(2)x^3(4^2)+(60)/(6)x^2(4)^3+(120)/(24)x^1(4)^4+(120)/(120)x^0(4)^5$

Now after solving the equation become

$(x-4)^5=x^5+20x^4+160x^3+640x^2+1280x+1024$

So, The after expanding the given statement with the binomial theorem the result output is

$(x-4)^5=x^5+20x^4+160x^3+640x^2+1280x+1024$

So, The binomial expansion of the $(x-4)^5$ is  $x^5+20x^4+160x^3+640x^2+1280x+1024$

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