Questions: Factor the polynomial. 6561-x^4 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. 6561-x^4 = (Type your answer in factored form. Simplify your answer.) B. The polynomial is prime.

Factor the polynomial.
6561-x^4

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. 6561-x^4 =
(Type your answer in factored form. Simplify your answer.)
B. The polynomial is prime.

Transcript text: Factor the polynomial. \[ 6561-x^{4} \] Select the correct choice below and, if hecessary, fill in the answer box to complete your choice. A. $6561-\mathrm{x}^{4}=$ $\square$ (Type your answer in factored form. Simplify your answer.) B. The polynomial is prime.

Solution

Solution Steps

To factor the polynomial $6561 - x^4$, we recognize it as a difference of squares. The expression can be rewritten as $(81)^2 - (x^2)^2$. We can apply the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$, to factor it further.

Solution Approach

Recognize the expression as a difference of squares: $6561 - x^4 = (81)^2 - (x^2)^2$.
Apply the difference of squares formula: $(a^2 - b^2) = (a - b)(a + b)$.
Factor the resulting expressions further if possible.

Step 1: Recognize the Expression

The polynomial $6561 - x^4$ can be identified as a difference of squares, which can be expressed as: \[ (81)^2 - (x^2)^2 \]

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula $a^2 - b^2 = (a - b)(a + b)$, we can factor the expression: \[ 6561 - x^4 = (81 - x^2)(81 + x^2) \]

Step 3: Further Factorization

The term $81 - x^2$ can also be factored further as another difference of squares: \[ 81 - x^2 = (9 - x)(9 + x) \] Thus, the complete factorization of the original polynomial is: \[ 6561 - x^4 = -(x - 9)(x + 9)(x^2 + 81) \]