Factor the following expression. \( 30-t^{10}-t^{20} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( 30-t^{10}-t^{20}=\square \) (Simplify your answer. Type your answer in factored form.) B. The given polynomial is not factorable.
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To factor the expression \( 30 - t^{10} - t^{20} \), we can rearrange it as \( -t^{20} - t^{10} + 30 \). This is equivalent to \( - (t^{20} + t^{10} - 30) \). We can treat this as a quadratic in terms of \( t^{10} \). Letting \( u = t^{10} \), we rewrite the expression as \( - (u^2 + u - 30) \). The quadratic \( u^2 + u - 30 \) factors into \( (u - 5)(u + 6) \) because \( -5 + 6 = 1 \) and \( -5 \cdot 6 = -30 \). Substituting back in for \( u \) gives us \( - (t^{10} - 5)(t^{10} + 6) \). So, the factored form is \( 30 - t^{10} - t^{20} = - (t^{10} - 5)(t^{10} + 6) \). In summary, the correct choice is: A. \( 30 - t^{10} - t^{20} = - (t^{10} - 5)(t^{10} + 6) \)
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