Expand Using the Binomial Theorem (x-5)^2

Problem

(x−5)2

Solution

1. Identify the terms and the exponent for the binomial expansion (a+b)n where a=x b=−5 and n=2

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 formula for k=0,1,2

(x−5)2=(2/0)x2*(−5)0+(2/1)x1*(−5)1+(2/2)x0*(−5)2)))

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

(2/0)=1)

(2/1)=2)

(2/2)=1)

5. Simplify each term by performing the multiplication and exponentiation.

1⋅x2⋅1=x2

2⋅x⋅(−5)=−10*x

1⋅1⋅25=25

6. Combine the terms to find the final expanded polynomial.

x2−10*x+25

Final Answer

(x−5)2=x2−10*x+25

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