If $x=-5+2 \sqrt{-4}$, find the value of $x^4+9 x^3+35 x^2-x+4$

Simplify the complex number: x = − 5 + 4 i .
Find a quadratic factor: ( x − ( − 5 + 4 i )) ( x − ( − 5 − 4 i )) = x 2 + 10 x + 41 .
Divide the polynomial: P ( x ) = ( x 2 + 10 x + 41 ) ( x 2 − x − 6 ) + 23 x + 250 .
Substitute x = − 5 + 4 i into the remainder: 23 ( − 5 + 4 i ) + 250 = 135 + 92 i .
The final answer is 135 + 92 i ​ .

Explanation

Simplify the complex number We are given the complex number x = − 5 + 2 − 4 ​ and asked to find the value of the polynomial x 4 + 9 x 3 + 35 x 2 − x + 4 . First, we simplify the expression for x . Since − 4 ​ = 2 i , we have x = − 5 + 4 i .

Find a quadratic factor Let P ( x ) = x 4 + 9 x 3 + 35 x 2 − x + 4 . Since the coefficients of P ( x ) are real, if x = − 5 + 4 i is a root of some polynomial with real coefficients, then its complex conjugate x = − 5 − 4 i is also a root. Thus, we can find a quadratic factor of P ( x ) by considering ( x − ( − 5 + 4 i )) ( x − ( − 5 − 4 i )) .

Calculate the quadratic factor Expanding this, we get ( x − ( − 5 + 4 i )) ( x − ( − 5 − 4 i )) = ( x + 5 − 4 i ) ( x + 5 + 4 i ) = (( x + 5 ) − 4 i ) (( x + 5 ) + 4 i ) = ( x + 5 ) 2 − ( 4 i ) 2 = x 2 + 10 x + 25 − 16 i 2 = x 2 + 10 x + 25 + 16 = x 2 + 10 x + 41. Thus, x 2 + 10 x + 41 is a quadratic factor.

Divide the polynomial Now we perform polynomial long division to divide P ( x ) = x 4 + 9 x 3 + 35 x 2 − x + 4 by x 2 + 10 x + 41 . We have


x 4 + 9 x 3 + 35 x 2 − x + 4 = ( x 2 + 10 x + 41 ) ( x 2 − x ) + ( − 6 x 2 − x + 4 ) = ( x 2 + 10 x + 41 ) ( x 2 − x − 6 ) + ( 23 x + 250 ) .
So, P ( x ) = ( x 2 + 10 x + 41 ) ( x 2 − x − 6 ) + 23 x + 250 .
Since x = − 5 + 4 i is a root of x 2 + 10 x + 41 , we have x 2 + 10 x + 41 = 0 . Therefore, P ( x ) = 23 x + 250 .

Substitute x into the remainder Substituting x = − 5 + 4 i into 23 x + 250 , we get 23 ( − 5 + 4 i ) + 250 = − 115 + 92 i + 250 = 135 + 92 i .

Final Answer Therefore, the value of x 4 + 9 x 3 + 35 x 2 − x + 4 when x = − 5 + 4 i is 135 + 92 i .


Examples
Polynomials are used to model curves and relationships in various fields such as physics, engineering, computer graphics, and economics. For example, in physics, projectile motion can be modeled using quadratic polynomials. In engineering, polynomials are used to design curves for roads and bridges. Understanding how to evaluate polynomials with complex numbers allows engineers and scientists to analyze systems with oscillating or damped behavior, which are common in electrical circuits and mechanical vibrations. This problem demonstrates how to evaluate a polynomial at a complex value, which is a fundamental skill in these applications.

Answered by GinnyAnswer | 2025-07-05