SOLUTION: 8) Use the Rational Zero Theorem to list all possible rational zeros for the given function f(x) = - 2x ^ 3 + 3x ^ 2 - 4x + 8

Algebra ->  Equations -> SOLUTION: 8) Use the Rational Zero Theorem to list all possible rational zeros for the given function f(x) = - 2x ^ 3 + 3x ^ 2 - 4x + 8      Log On


   

Question 1168599: 8) Use the Rational Zero Theorem to list all possible rational zeros for the given function
f(x) = - 2x ^ 3 + 3x ^ 2 - 4x + 8
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Use the Rational Zero Theorem to list all possible rational zeros for the given function
f%28x%29+=+-2x%5E3+%2B+3x%5E2+-4x+%2B+8

Since all coefficients are integers, we can apply the rational zeros theorem.
The trailing coefficient (coefficient of the constant term) is 8.
Find its factors (with plus and minus): ±1,±2,±4,±8. These are the possible values for p.
The leading coefficient (coefficient of the term with the highest degree) is -2.
Find its factors (with plus and minus): ±1,±2. These are the possible values for q.
Find all possible values of p%2Fq:
±1%2F1,±1%2F2,±2%2F1,±2%2F2,±4%2F1,±4%2F2,±8%2F1,±8%2F2.
Simplify and remove duplicates (if any), these are highlight%28possible%29 rational roots: ±1,±1%2F2,±2,±4,±8


Next, check the possible roots:
if highlight%28a%29 is a root of the polynomial f%28x%29, the remainder from the division of f%28x%29 by highlight%28x-a%29 should equal highlight%280%29.
Check 1: divide -2x%5E3%2B3x%5E2-4x%2B8 by x-1.The quotient is -2x%5E2%2Bx-3 and the remainder is 5 (use the synthetic division calculator to see the steps).
Check -1: divide -2x%5E3%2B3x%5E2-4x%2B8 by x%2B1.The quotient is -2x%5E2%2B5x-9 and the remainder is 17 (use the synthetic division calculator to see the steps).
Check 1%2F2: divide -2x%5E3%2B3x%5E2-4x%2B8 by x-1%2F2.The quotient is -2x%5E2%2B2x%E2%88%923 and the remainder is 13%2F2 (use the synthetic division calculator to see the steps).
Check -1%2F2: divide -2x%5E3%2B3x%5E2-4x%2B8 by x%2B1%2F2.The quotient is -2x%5E2%2B4x-6 and the remainder is 11 (use the synthetic division calculator to see the steps).
Check 2: divide -2x%5E3%2B3x%5E2-4x%2B8 by x-2.The quotient is -2x%5E2-x-6 and the remainder is -4 (use the synthetic division calculator to see the steps).
Check -2: divide -2x%5E3%2B3x%5E2-4x%2B8 by x%2B2.The quotient is -2x%5E2%2B7x-18 and the remainder is 44 (use the synthetic division calculator to see the steps).
Check 4: divide -2x%5E3%2B3x%5E2-4x%2B8 by x-4.The quotient is -2x%5E2-5x-24 and the remainder is -88 (use the synthetic division calculator to see the steps).
Check -4: divide -2x%5E3%2B3x%5E2-4x%2B8 by x%2B4.The quotient is -2x%5E2%2B11x-48 and the remainder is 200+(use the synthetic division calculator to see the steps).
Check 8: divide -2x%5E3%2B3x%5E2-4x%2B8 by x-8.The quotient is -2x%5E2-13x-108 and the remainder is -856 (use the synthetic division calculator to see the steps).
Check -8: divide -2x%5E3%2B3x%5E2-4x%2B8 by x%2B8.The quotient is -2x%5E2%2B19x-156 and the remainder is 1256 (use the synthetic division calculator to see the steps).

so, none of these possible rational roots ±1,±1%2F2,±2,±4,±8 are real roots of given function because long division by them does not gives us reminder highlight%280%29