Factor 2x^3-9x^2-6x+5

Problem

2*x^3−9*x^2−6*x+5

Solution

1. Identify potential rational roots using the Rational Root Theorem, which suggests testing factors of the constant term 5 divided by factors of the leading coefficient 2

2. Test the value x=5 using synthetic division or direct substitution.

3. Evaluate the polynomial at x=5

2*(5)^3−9*(5)^2−6*(5)+5=250−225−30+5=0

4. Divide the polynomial by the factor (x−5) using synthetic division to find the remaining quadratic factor.

(2*x^3−9*x^2−6*x+5)÷(x−5)=2*x^2+x−1

5. Factor the resulting quadratic expression 2*x^2+x−1 by splitting the middle term or using the quadratic formula.

2*x^2+x−1=(2*x−1)*(x+1)

6. Combine all factors to write the completely factored form of the original cubic polynomial.

Final Answer

2*x^3−9*x^2−6*x+5=(x−5)*(2*x−1)*(x+1)

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