Question
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.
Ask by Martinez Parry. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(30 - t^{10} - t^{20} = (5 - t^{10})(6 + t^{10})\)
Therefore, the correct choice is:
A. \(30 - t^{10} - t^{20} = (5 - t^{10})(6 + t^{10})\)
Solution
Factor the expression \( 30-t^{10}-t^{20} \).
Factor the expression by following steps:
- step0: Factor:
\(30-t^{10}-t^{20}\)
- step1: Rewrite the expression:
\(30+\left(5-6\right)t^{10}-t^{20}\)
- step2: Calculate:
\(30+5t^{10}-6t^{10}-t^{20}\)
- step3: Rewrite the expression:
\(5\times 6+5t^{10}-t^{10}\times 6-t^{10}\times t^{10}\)
- step4: Factor the expression:
\(5\left(6+t^{10}\right)-t^{10}\left(6+t^{10}\right)\)
- step5: Factor the expression:
\(\left(5-t^{10}\right)\left(6+t^{10}\right)\)
The expression \(30-t^{10}-t^{20}\) can be factored as \((5-t^{10})(6+t^{10})\).
Therefore, the correct choice is:
A. \(30-t^{10}-t^{20}=(5-t^{10})(6+t^{10})\)
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
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. \( 30 - t^{10} - t^{20} = - (t^{10} - 5)(t^{10} + 6) \)
