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Answer :
Final answer:
To factor the polynomial P(x) = 6x5 – 35x4 + 39x3 + 59x2 – 57x – 36, we find that x = 1 is a root and divide P(x) by (x - 1) to obtain the factored form.
Explanation:
To factor the polynomial P(x) = 6x5 – 35x4 + 39x3 + 59x2 – 57x – 36, we start by looking for any common factors among the terms. In this case, there are no common factors.
Next, we can try to find the roots of the polynomial by using the Rational Root Theorem. The potential rational roots of P(x) are the factors of the constant term (in this case, ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36). By testing these values, we find that x = 1 is a root of P(x).
Using synthetic division or long division, we divide P(x) by (x - 1) to obtain a quotient polynomial. In this case, the quotient polynomial is Q(x) = 6x4 + x3 + 40x2 + 99x + 36.
The factored form of P(x) is then (x - 1)(6x4 + x3 + 40x2 + 99x + 36).
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