Find the Roots (Zeros) f(x)=x^3+12x^2+21x+10

Problem

ƒ(x)=x3+12*x2+21*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

2. Test the value x=−1 using synthetic division or direct substitution.

3. Substitute x=−1 into the function: ƒ*(−1)=(−1)3+12*(−1)2+21*(−1)+10=−1+12−21+10=0

4. Factor out (x+1) from the polynomial using synthetic division to find the remaining quadratic factor.

5. Divide the polynomial: (x3+12*x2+21*x+10)÷(x+1)=x2+11*x+10

6. Factor the resulting quadratic expression x2+11*x+10 by finding two numbers that multiply to 10 and add to 11

7. Identify the factors as (x+10) and (x+1)

8. Set each factor to zero to find all roots: x+1=0 and x+10=0

Final Answer

x=−1,−10

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