Factor x^4-8x^2+16

Problem

x^4−8*x^2+16

Solution

1. Identify the expression as a trinomial in the form of a perfect square, specifically a^2−2*a*b+b^2

2. Determine the square roots of the first and last terms, where a=√(,x^4)=x^2 and b=√(,16)=4

3. Verify the middle term by calculating 2*a*b=2*(x^2)*(4)=8*x^2 which matches the given expression.

4. Apply the perfect square trinomial formula (a−b)^2 to get (x^2−4)^2

5. Recognize that the inner expression x^2−4 is a difference of squares, which factors into (x−2)*(x+2)

6. Substitute this factorization back into the squared expression to get ((x−2)*(x+2))^2

7. Distribute the exponent to each factor.

Final Answer

x^4−8*x^2+16=(x−2)^2*(x+2)^2

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