Find the Roots/Zeros Using the Rational Roots Test 2x^4-17x^3+35x^2+9x-45

Problem

2*x4−17*x3+35*x2+9*x−45

Solution

1. Identify the constant term (a_0)=−45 and the leading coefficient (a_n)=2

2. List all possible factors of the constant term p∈{±1,±3,±5,±9,±15,±45} and all possible factors of the leading coefficient q∈{1,2}

3. Determine the set of possible rational roots p/q by dividing each factor of p by each factor of q

p/q∈{±1,±3,±5,±9,±15,±45,±1/2,±3/2,±5/2,±9/2,±15/2,±45/2}

4. Test values using synthetic division or substitution. Testing x=3

2*(3)4−17*(3)3+35*(3)2+9*(3)−45=162−459+315+27−45=0

So, x=3 is a root.

5. Perform synthetic division by (x−3) to reduce the polynomial to 2*x3−11*x2+2*x+15

6. Test values for the reduced polynomial. Testing x=3 again:

2*(3)3−11*(3)2+2*(3)+15=54−99+6+15≠0

Testing x=−1

2*(−1)3−11*(−1)2+2*(−1)+15=−2−11−2+15=0

So, x=−1 is a root.

7. Perform synthetic division by (x+1) on the reduced polynomial to get 2*x2−13*x+15

8. Factor the remaining quadratic equation 2*x2−13*x+15=0

(2*x−3)*(x−5)=0

Setting each factor to zero gives x=3/2 and x=5

Final Answer

2*x4−17*x3+35*x2+9*x−45=0⇒x∈{−1,3/2,3,5}

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