Find the Roots (Zeros) f(x)=x^3+5x^2-73x-77

Problem

ƒ(x)=x3+5*x2−73*x−77

Solution

1. Identify potential rational roots using the Rational Root Theorem, which suggests testing factors of the constant term −77 divided by factors of the leading coefficient 1

2. Test the value x=−1 by substituting it into the function.

ƒ*(−1)=(−1)3+5*(−1)2−73*(−1)−77

ƒ*(−1)=−1+5+73−77=0

3. Divide the polynomial by the factor (x+1) using synthetic division or long division to find the remaining quadratic factor.

(x3+5*x2−73*x−77)÷(x+1)=x2+4*x−77

4. Factor the resulting quadratic expression x2+4*x−77 by finding two numbers that multiply to −77 and add to 4

x2+4*x−77=(x+11)*(x−7)

5. Set each factor to zero to determine all roots of the polynomial.

x+1=0⇒x=−1

x+11=0⇒x=−11

x−7=0⇒x=7

Final Answer

x={−11,−1,7}

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