If $(x+1)$ and $(x-1)$ are factors of $x^4+a x^3-3 x^2+2 x+b$, find the values of $a$ and $b$.

The values of a and b are found to be a = − 2 and b = 2 by substituting the roots of the factors into the polynomial and solving the resulting equations.
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Answered by Anonymous | 2025-07-04

Substitute x = 1 and x = − 1 into the polynomial x 4 + a x 3 − 3 x 2 + 2 x + b and set the results to 0, since ( x + 1 ) and ( x − 1 ) are factors.
Obtain two equations: a + b = 0 and − a + b = 4 .
Solve the system of equations to find the values of a and b .
The solution is a = − 2 and b = 2 , so the final answer is a = − 2 , b = 2 ​ .

Explanation

Understanding the Problem We are given that ( x + 1 ) and ( x − 1 ) are factors of the polynomial P ( x ) = x 4 + a x 3 − 3 x 2 + 2 x + b . Our goal is to find the values of a and b . Since ( x + 1 ) and ( x − 1 ) are factors of P ( x ) , it means that P ( 1 ) = 0 and P ( − 1 ) = 0 . This is because if a polynomial P ( x ) has a factor ( x − c ) , then P ( c ) = 0 .

Substituting x=1 First, let's substitute x = 1 into P ( x ) :


P ( 1 ) = ( 1 ) 4 + a ( 1 ) 3 − 3 ( 1 ) 2 + 2 ( 1 ) + b = 1 + a − 3 + 2 + b = a + b = 0
So, we have the equation:
a + b = 0

Substituting x=-1 Next, let's substitute x = − 1 into P ( x ) :

P ( − 1 ) = ( − 1 ) 4 + a ( − 1 ) 3 − 3 ( − 1 ) 2 + 2 ( − 1 ) + b = 1 − a − 3 − 2 + b = − a + b − 4 = 0
So, we have the equation:
− a + b = 4

Solving the System of Equations Now we have a system of two equations with two variables a and b :

{ a + b = 0 − a + b = 4 ​
We can solve this system of equations. From the first equation, we have a = − b . Substituting this into the second equation, we get:
− ( − b ) + b = 4
b + b = 4
2 b = 4
b = 2

Finding the Values of a and b Now that we have b = 2 , we can find a using the equation a = − b :

a = − 2
So, the values are a = − 2 and b = 2 .

Final Answer Therefore, the values of a and b are − 2 and 2 respectively.

a = − 2 , b = 2
Examples
Understanding factors of polynomials is crucial in various fields, such as engineering and physics, where polynomial equations are used to model physical phenomena. For instance, in control systems, the stability of a system can be determined by analyzing the roots (or factors) of a characteristic polynomial. Similarly, in structural analysis, polynomial equations are used to describe the deformation of structures under load. By finding the factors of these polynomials, engineers can predict the behavior of the system or structure and design accordingly. This ensures safety and efficiency in real-world applications.

Answered by GinnyAnswer | 2025-07-04