Solve for x 5x^3+7x^2-11x+3=0

Problem

5*x3+7*x2−11*x+3=0

Solution

1. Identify possible rational roots using the Rational Root Theorem, which suggests testing factors of the constant term 3 divided by factors of the leading coefficient 5

2. Test the value x=1 by substituting it into the polynomial: 5*(1)3+7*(1)2−11*(1)+3=5+7−11+3=4 Since this is not zero, test x=−3

3. Substitute x=−3 into the equation: 5(-27) + 7(9) - 11(-3) + 3 = -135 + 63 + 33 + 3 = -36.T*e*s(t) = \frac{3}{5}$.

4. Substitute x=3/5 into the equation: 5*(27/125)+7*(9/25)−11*(3/5)+3=27/25+63/25−165/25+75/25=0 Thus, (5*x−3) is a factor.

5. Divide the polynomial 5*x3+7*x2−11*x+3 by (5*x−3) using synthetic or long division to find the remaining quadratic factor.

6. Result of the division is x2+2*x−1

7. Solve the quadratic equation x2+2*x−1=0 using the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a)

8. Calculate the roots: x=(−2±√(,4−4*(1)*(−1)))/2=(−2±√(,8))/2=(−2±2√(,2))/2=−1±√(,2)

Final Answer

x=3/5,−1+√(,2),−1−√(,2)

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