Questions: Factor the binomial completely. x^8-81 x^4 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. x^8-81 x^4= (Factor completely.) B. The polynomial is prime.

Solution Steps

To factor the binomial \( x^8 - 81x^4 \) completely, we can use the following steps:

1. Recognize that \( x^8 - 81x^4 \) can be factored by treating it as a difference of squares.
2. Rewrite \( x^8 \) as \( (x^4)^2 \) and \( 81x^4 \) as \( (9x^2)^2 \).
3. Apply the difference of squares formula: \( a^2 - b^2 = (a - b)(a + b) \).
4. Continue factoring any resulting expressions that can be further factored.

The given expression is \( x^8 - 81x^4 \). We can recognize this as a difference of squares: \[ x^8 - 81x^4 = (x^4)^2 - (9x^2)^2 \]

Using the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \), we can factor the expression: \[ (x^4)^2 - (9x^2)^2 = (x^4 - 9x^2)(x^4 + 9x^2) \]

We can further factor \( x^4 - 9x^2 \) as another difference of squares: \[ x^4 - 9x^2 = x^2(x^2 - 9) = x^2(x - 3)(x + 3) \] The term \( x^4 + 9x^2 \) cannot be factored further over the real numbers.

Final Answer