The polynomial equation $x^3+x^2=-9 x-9$ has complex roots $\pm 3 i$. What is the other root?
Answer (2)
The remaining root of the polynomial equation x 3 + x 2 + 9 x + 9 = 0 , given the complex roots ± 3 i , is − 1 .
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Rewrite the given polynomial equation in standard form: x 3 + x 2 + 9 x + 9 = 0 .
Recognize that since ± 3 i are roots, ( x 2 + 9 ) is a factor of the polynomial.
Factor the polynomial by grouping: ( x 2 + 9 ) ( x + 1 ) = 0 .
Determine the remaining root from the factor ( x + 1 ) , which is x = − 1 . The final answer is − 1 .
Explanation
Problem Analysis We are given the polynomial equation x 3 + x 2 = − 9 x − 9 and told that it has complex roots ± 3 i . Our goal is to find the remaining root.
Rewrite the Equation First, let's rewrite the equation in standard form by moving all terms to one side: x 3 + x 2 + 9 x + 9 = 0
Identify a Factor Since we know that 3 i and − 3 i are roots of the equation, we know that ( x − 3 i ) and ( x + 3 i ) are factors of the polynomial. Therefore, their product, ( x − 3 i ) ( x + 3 i ) = x 2 − ( 3 i ) 2 = x 2 − ( − 9 ) = x 2 + 9 , must also be a factor of the polynomial.
Find the Remaining Factor Now, we can perform polynomial division to find the remaining factor. We divide x 3 + x 2 + 9 x + 9 by x 2 + 9 . Alternatively, we can try to factor by grouping: x 3 + x 2 + 9 x + 9 = x 2 ( x + 1 ) + 9 ( x + 1 ) = ( x 2 + 9 ) ( x + 1 )
Determine the Roots From the factored form ( x 2 + 9 ) ( x + 1 ) = 0 , we can see that the roots are the solutions to x 2 + 9 = 0 and x + 1 = 0 . The solutions to x 2 + 9 = 0 are x = ± 3 i , which we already knew. The solution to x + 1 = 0 is x = − 1 .
State the Answer Therefore, the other root is − 1 .
Examples
Polynomial equations can model various real-world phenomena, such as the trajectory of a projectile or the behavior of electrical circuits. In circuit analysis, the roots of a characteristic equation determine the stability and response of the circuit. Finding the roots helps engineers design circuits that behave predictably and meet specific performance criteria. For example, the equation x 3 + x 2 + 9 x + 9 = 0 could represent a simplified model of a circuit, where the roots indicate critical frequencies or damping factors.
