Question

factor ( x^4 - 18x^2 + 81 ) completely. all factors in your answer should have integer coefficients.

Explanation:

Step1: Recognize the quadratic form

Let \( y = x^2 \), then the expression \( x^4 - 18x^2 + 81 \) becomes \( y^2 - 18y + 81 \).

Step2: Factor the quadratic in y

We need to find two numbers that multiply to 81 and add up to -18. Those numbers are -9 and -9. So, \( y^2 - 18y + 81=(y - 9)^2 \).

Step3: Substitute back \( y = x^2 \)

Substituting back, we get \( (x^2 - 9)^2 \).

Step4: Factor \( x^2 - 9 \)

Notice that \( x^2 - 9 \) is a difference of squares, which factors as \( (x - 3)(x + 3) \).

Step5: Write the final factored form

So, \( (x^2 - 9)^2 = [(x - 3)(x + 3)]^2=(x - 3)^2(x + 3)^2 \).

Answer:

\((x - 3)^2(x + 3)^2\)