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Possible rational roots: ±1,±5.

Actual rational roots: −1,5.

Step-by-step explanation:

The trailing coefficient (coefficient of the constant term) is −5.

Find its factors (with plus and minus): ±1,±5. These are the possible values for p.

The leading coefficient (coefficient of the term with the highest degree) is 1.

Find its factors (with plus and minus): ±1. These are the possible values for q.

Find all possible values of pq: ±11,±51.

Simplify and remove duplicates (if any), these are possible rational roots: ±1,±5.

Next, check the possible roots: if a is a root of the polynomial P(x), the remainder from the division of P(x) by x−a should equal 0.

Check 1: divide x3−3x2−9x−5 by x−1.The quotient is x2−2x−11 and the remainder is −16  

Check −1: divide x3−3x2−9x−5 by x+1.The quotient is x2−4x−5 and the remainder is 0  

So, −1 is a root.

Check 5: divide x3−3x2−9x−5 by x−5.The quotient is x2+2x+1 and the remainder is 0  

So, 5 is a root.

Check −5: divide x3−3x2−9x−5 by x+5.The quotient is x2−8x+31 and the remainder is −160  

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step-by-step explanation:

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