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Use synthetic division and the given factor to completely factor the polynomial.
x3 – x2 – 24x – 36; (x – 6)
To factor the polynomial \(x^3 - x^2 - 24x - 36\) using synthetic division with the factor (x - 6), follow these steps:
1. Write down the coefficients of the polynomial: 1, -1, -24, -36
2. Write down the factor (x - 6) for the synthetic division.
3. Bring down the first coefficient (1) to start the process.
4. Multiply the number at the bottom by the number you just brought down, and write the result below the next coefficient.
5. Add the two numbers together, and repeat the process until you reach the end.
6. The numbers in the last row are the coefficients of the quotient polynomial.
7. The constant at the very right is the remainder.
Therefore, the factored form of the polynomial is:
\(x^3 - x^2 - 24x - 36 = (x - 6)(x^2 + 5x + 6) = (x - 6)(x + 2)(x + 3)\)