Find the Roots (Zeros) f(x)=x^4-x^3-7x^2+5x+10

Problem

ƒ(x)=x4−x3−7*x2+5*x+10

Solution

1. Identify potential rational roots using the Rational Root Theorem, which suggests testing factors of the constant term 10 divided by factors of the leading coefficient 1 The possible roots are ±1,±2,±5,±10

2. Test the value x=−1 using synthetic division or direct substitution. Since ƒ*(−1)=(−1)4−(−1)3−7*(−1)2+5*(−1)+10=1+1−7−5+10=0 then (x+1) is a factor.

3. Divide the polynomial by (x+1) to obtain the depressed polynomial x3−2*x2−5*x+10

4. Factor the resulting cubic polynomial x3−2*x2−5*x+10 by grouping. Group the first two terms and the last two terms: x2*(x−2)−5*(x−2)

5. Extract the common binomial factor (x−2) which results in (x−2)*(x2−5)

6. Solve for the remaining roots by setting each factor to zero. The factors are (x+1) (x−2) and (x2−5)

7. Calculate the roots of x2−5=0 by adding 5 to both sides and taking the square root, yielding x=±√(,5)

Final Answer

x=−1,2,√(,5),−√(,5)

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