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)*b^k)

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

(4/0)x^4*(−4)^0+(4/1)x^3*(−4)^1+(4/2)x^2*(−4)^2+(4/3)x^1*(−4)^3+(4/4)x^0*(−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⋅x^4⋅1+4⋅x^3⋅(−4)+6⋅x^2⋅16+4⋅x⋅(−64)+1⋅1⋅256

6. Combine the results to find the final polynomial:

x^4−16*x^3+96*x^2−256*x+256

Final Answer

(x−4)^4=x^4−16*x^3+96*x^2−256*x+256

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