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Answer
\( t^{2}(r - s) - r + s = (t - 1)(t + 1)(r - s) \)
Solution
To factorize the expression \( t^{2}(r - s) - r + s \) fully, follow these steps:
1. **Rearrange the Expression:**
\[
t^{2}(r - s) - r + s = t^{2}(r - s) - (r - s)
\]
2. **Factor Out the Common Term \((r - s)\):**
\[
t^{2}(r - s) - (r - s) = (t^{2} - 1)(r - s)
\]
3. **Factorize \( t^{2} - 1 \) as a Difference of Squares:**
\[
t^{2} - 1 = (t - 1)(t + 1)
\]
4. **Combine All Factors:**
\[
(t^{2} - 1)(r - s) = (t - 1)(t + 1)(r - s)
\]
**Final Factored Form:**
\[
(t - 1)(t + 1)(r - s)
\]
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Simplify this solution Bonus Knowledge
To factor \( t^{2}(r-s)-r+s \), first, we can rearrange it: \[ t^{2}(r-s) + (s-r) \] Next, we'll factor out the common term \( (r-s) \): \[ (r-s)(t^{2} - 1) \] Now, recognizing that \( t^{2} - 1 \) is a difference of squares, we can factor it further: \[ (r-s)(t-1)(t+1) \] So, the final fully factored form of the expression \( t^{2}(r-s)-r+s \) is: \[ (r-s)(t-1)(t+1) \]
