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Answer :
Final answer:
To calculate P(7), synthetic division is used by taking the coefficients of P(x) and using 7 as a synthetic number to progressively add and multiply until reaching the final result which is P(7).
Explanation:
To calculate P(7) for the given polynomial P(x) = 6x⁷ − 39x⁶ + 24x⁵ − 305x⁴ − 60x³ − 69x² + 14x − 144 using synthetic division, we follow these steps:
- Write down the coefficients of P(x): 6, -39, 24, -305, -60, -69, 14, -144.
- Bring down the 6 and multiply it by 7 (our synthetic number), which is 42, and add this to -39, getting 3.
- Continue this process for each coefficient. Multiply the last result by 7 and add it to the next coefficient.
- After the last multiplication and addition, the final number achieved is P(7).
The process involves synthetic division which reduces computational complexity compared to the long division of polynomials and is ideal for evaluating polynomials at a given point.
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