Find all the roots of: F(x)= x^3-2x^2-2x-3

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jgeertz | College Teacher | (Level 1) Associate Educator

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If a polynomial function has integer coefficients, then every rational zero will have the form p/q where p is a factor of the constant and q is a factor of the leading coefficient.

p=3

q=1

The possible roots of the polynomial function are every combination of +- p/q

+-1, +-3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is 0, which means it is a root.

(3)^3 - 2(3)^2 - 2(3) -3

Simplify the expression. It is equal to 0 so x=3 is a root of the polynomial.

Since 3 is a root, divide the polynomial by (x-3) to find the quotient polynomial. This polynomial can be used to find the remaining roots.

x^3-2x^2-2x-3/x-3

After completing division, we are left with x^2+x+1.

Set this equation to zero.

x^2 +x+1 =0

Use the quadratic formula to find the solutions. a=1, b=1, c=1

x= -1+i (sq.rt3)/2 and  -1-i (sq.rt3)/2

 

These are the roots of the polynomial x^2-2x^2-2x-3.

x=3  

x= -1+i(sq.rt3)/2  

x= -1-i(sq.rt3)/2

 

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