If [tex]$P(x) = Q(x) \cdot (x - a)$[/tex], then a is a solution. Express [tex]$P(x) = 2x^3 - 5x^2 + 5x - 6$[/tex] in the form [tex]$P(x) = Q(x) \cdot (x - a)$[/tex]
Answer (2)
Find a root 'a' of the polynomial P(x) by testing divisors of the constant term.
Perform polynomial division to divide P(x) by (x-a) and find the quotient Q(x).
Express P(x) in the form P(x) = Q(x) \cdot (x-a).
The final answer is P ( x ) = ( 2 x 2 − x + 3 ) ( x − 2 ) .
Explanation
Understanding the Problem We are given the polynomial P ( x ) = 2 x 3 − 5 x 2 + 5 x − 6 and we want to express it in the form P ( x ) = Q ( x ) \t ⋅ ( x − a ) , where a is a root of P ( x ) . This means we need to find a value a such that P ( a ) = 0 .
Finding a Root To find a root of P ( x ) , we can test integer divisors of the constant term, which is -6. The divisors of -6 are ± 1 , ± 2 , ± 3 , ± 6 . Let's test these values:
For x = 1 , P ( 1 ) = 2 ( 1 ) 3 − 5 ( 1 ) 2 + 5 ( 1 ) − 6 = 2 − 5 + 5 − 6 = − 4 = 0 .
For x = 2 , P ( 2 ) = 2 ( 2 ) 3 − 5 ( 2 ) 2 + 5 ( 2 ) − 6 = 2 ( 8 ) − 5 ( 4 ) + 10 − 6 = 16 − 20 + 10 − 6 = 0 .
Polynomial Division Since P ( 2 ) = 0 , x = 2 is a root of P ( x ) . Therefore, we can write P ( x ) = Q ( x ) ⋅ ( x − 2 ) for some polynomial Q ( x ) . To find Q ( x ) , we can perform polynomial division of P ( x ) by ( x − 2 ) .
Determining the Quotient Dividing 2 x 3 − 5 x 2 + 5 x − 6 by ( x − 2 ) , we get:
2x^2 - x + 3
____________________
x - 2 | 2x^3 - 5x^2 + 5x - 6
- (2x^3 - 4x^2)
____________________
-x^2 + 5x
- (-x^2 + 2x)
____________________
3x - 6
- (3x - 6)
____________________
0
So, Q ( x ) = 2 x 2 − x + 3 .
Expressing P(x) in the desired form Therefore, we can express P ( x ) as P ( x ) = ( 2 x 2 − x + 3 ) ( x − 2 ) .
Final Answer Thus, P ( x ) = ( 2 x 2 − x + 3 ) ( x − 2 ) .
Examples
Polynomial factorization is a fundamental concept in algebra and has practical applications in various fields. For instance, in engineering, when analyzing the stability of a system, we often need to find the roots of a characteristic polynomial. Factoring the polynomial helps in identifying these roots, which determine the system's stability. Similarly, in physics, solving equations of motion often involves finding the roots of a polynomial, and factorization simplifies this process. In computer graphics, polynomial curves are used to model shapes, and understanding their roots helps in rendering and manipulating these shapes effectively. Factoring polynomials allows us to understand the behavior and properties of these systems and models.
To express P ( x ) = 2 x 3 − 5 x 2 + 5 x − 6 in the form P ( x ) = Q ( x ) ⋅ ( x − a ) , we found that a root is a = 2 . After performing polynomial division, we determined that Q ( x ) = 2 x 2 − x + 3 . Thus, the final expression is P ( x ) = ( 2 x 2 − x + 3 ) ( x − 2 ) .
;
