The polynomial equation $x^3+x^2=-9 x-9$ has complex roots $\pm 3 i$. What is the other root?
Answer (2)
Rewrite the equation as x 3 + x 2 + 9 x + 9 = 0 .
Recognize that since ± 3 i are roots, ( x 2 + 9 ) is a factor.
Divide the polynomial by ( x 2 + 9 ) to find the remaining factor ( x + 1 ) .
The remaining root is x = − 1 , so 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, rewrite the equation in standard form by moving all terms to the left side: x 3 + x 2 + 9 x + 9 = 0
Identify a Quadratic Factor Since we know that 3 i and − 3 i are roots of the polynomial, we know that ( x − 3 i ) and ( x + 3 i ) are factors. Therefore, their product ( x − 3 i ) ( x + 3 i ) = x 2 − ( 3 i ) 2 = x 2 − ( − 9 ) = x 2 + 9 is also 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, since we know that the polynomial can be written as ( x 2 + 9 ) ( x + a ) for some a , we can expand this to x 3 + a x 2 + 9 x + 9 a . Comparing coefficients, we see that a = 1 , so the remaining factor is ( x + 1 ) .
Determine the Remaining Root Thus, the polynomial can be factored as ( x 2 + 9 ) ( x + 1 ) = 0 . The roots are x = ± 3 i and x = − 1 . The remaining root is therefore − 1 .
Final Answer The other root is − 1 .
Examples
Polynomial equations are used in various fields such as physics, engineering, and economics. For example, in physics, polynomial equations can be used to model the trajectory of a projectile or the oscillations of a pendulum. In engineering, they can be used to design bridges or analyze electrical circuits. In economics, they can be used to model supply and demand curves or to forecast economic growth. Understanding how to find the roots of polynomial equations is essential for solving these types of problems.
The remaining root of the polynomial equation is − 1 . This is found by factoring the polynomial with the known complex roots ± 3 i . The complete roots of the polynomial are 3 i , − 3 i , and − 1 .
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