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

Problem

ƒ(x)=x4+x3−3*x2−x+2

Solution

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

2. Test x=1 by substituting it into the function: ƒ(1)=1+1−3*(1)2−1+2=1+1−3−1+2=0 Since ƒ(1)=0 (x−1) is a factor.

3. Divide the polynomial by (x−1) using synthetic division or long division to find the quotient.

x4+x3−3*x2−x+2=(x−1)*(x3+2*x2−x−2)

4. Factor the resulting cubic polynomial x3+2*x2−x−2 by grouping.

x2*(x+2)−1*(x+2)=(x2−1)*(x+2)

5. Expand the difference of squares (x2−1) into (x−1)*(x+1)

6. Combine all factors to write the fully factored form of the function.

ƒ(x)=(x−1)*(x−1)*(x+1)*(x+2)

ƒ(x)=(x−1)2*(x+1)*(x+2)

7. Solve for x by setting each factor to zero: x−1=0 x+1=0 and x+2=0

Final Answer

x=1,−1,−2

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