Expand Using the Binomial Theorem (x-4)^4

Problem

(x−4)4

Solution

1. Identify the components of the binomial expression (a+b)n where a=x b=−4 and n=4

2. Apply the Binomial Theorem formula, which states (a+b)n=(∑_k=0^n)((n/k))*a(n−k)*bk)

3. Write the expansion by substituting the values into the summation for k=0,1,2,3,4

(4/0)x4*(−4)0+(4/1)x3*(−4)1+(4/2)x2*(−4)2+(4/3)x1*(−4)3+(4/4)x0*(−4)4)))))

4. Calculate the binomial coefficients using (n/k)=(n!)/(k!(n−k)!))

(4/0)=1,(4/1)=4,(4/2)=6,(4/3)=4,(4/4)=1)))))

5. Simplify each term by multiplying the coefficients and the powers of −4

1⋅x4⋅1+4⋅x3⋅(−4)+6⋅x2⋅16+4⋅x⋅(−64)+1⋅1⋅256

6. Combine the results to find the final polynomial:

x4−16*x3+96*x2−256*x+256

Final Answer

(x−4)4=x4−16*x3+96*x2−256*x+256

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