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Find the Roots (Zeros) p(x)=12x^3+16x^2-x-5

Problem

p(x)=12*x3+16*x2−x−5

Solution

  1. Identify possible rational roots using the Rational Root Theorem, which suggests testing factors of the constant term −5 divided by factors of the leading coefficient 12

  2. Test the value x=−1 by substituting it into the polynomial: p*(−1)=12*(−1)3+16*(−1)2−(−1)−5=−12+16+1−5=0

  3. Divide the polynomial by the factor (x+1) using synthetic division or long division to find the remaining quadratic factor.

  4. Perform the division: (12*x3+16*x2−x−5)÷(x+1)=12*x2+4*x−5

  5. Factor the resulting quadratic 12*x2+4*x−5 using the grouping method or the quadratic formula.

  6. Solve 12*x2+10*x−6*x−5=0 which factors as 2*x*(6*x+5)−1*(6*x+5)=(2*x−1)*(6*x+5)

  7. Set each factor to zero to find the remaining roots: 2*x−1=0⇒x=1/2 and 6*x+5=0⇒x=−5/6

Final Answer

x=−1,1/2,−5/6

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