Expand Using the Binomial Theorem (x-2)^4

Problem

(x−2)4

Solution

1. Identify the parameters for the Binomial Theorem formula (a+b)n=(∑_k=0^n)((n/k))*a(n−k)*bk) Here, a=x b=−2 and n=4

2. Write the expansion using the binomial coefficients (4/0),(4/1),(4/2),(4/3),(4/4))))))

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

3. Calculate the binomial coefficients using Pascal's Triangle or the formula (n/k)=(n!)/(k!(n−k)!))

(4/0)=1)

(4/1)=4)

(4/2)=6)

(4/3)=4)

(4/4)=1)

4. Substitute the coefficients and powers of −2 back into the expression.

(x−2)4=1*(x4)*(1)+4*(x3)*(−2)+6*(x2)*(4)+4*(x)*(−8)+1*(1)*(16)

5. Simplify each term by multiplying the constants.

(x−2)4=x4−8*x3+24*x2−32*x+16

Final Answer

(x−2)4=x4−8*x3+24*x2−32*x+16

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