Select all polynomials that are divisible by $(x-1)$.

Choose all answers that apply:
A. $A(x)=3 x^3+2 x^2-x$
B. $B(x)=5 x^3-4 x^2-x$
C. $C(x)=2 x^3-3 x^2+2 x-1$
D. $D(x)=x^3+2 x^2+3 x+2$

Evaluate each polynomial at x = 1 .
A ( 1 ) = 4 , so A ( x ) is not divisible by ( x − 1 ) .
B ( 1 ) = 0 , so B ( x ) is divisible by ( x − 1 ) .
C ( 1 ) = 0 , so C ( x ) is divisible by ( x − 1 ) .
D ( 1 ) = 8 , so D ( x ) is not divisible by ( x − 1 ) .

The polynomials divisible by ( x − 1 ) are B ( x ) and C ( x ) . B , C ​
Explanation

Understanding the Problem We are given four polynomials: A ( x ) = 3 x 3 + 2 x 2 − x , B ( x ) = 5 x 3 − 4 x 2 − x , C ( x ) = 2 x 3 − 3 x 2 + 2 x − 1 , and D ( x ) = x 3 + 2 x 2 + 3 x + 2 . We need to determine which of these polynomials are divisible by ( x − 1 ) . A polynomial P ( x ) is divisible by ( x − 1 ) if and only if P ( 1 ) = 0 .

Solution Plan For each polynomial, we will substitute x = 1 into the polynomial and check if the result is 0.

Evaluating A(1) Let's evaluate A ( 1 ) : A ( 1 ) = 3 ( 1 ) 3 + 2 ( 1 ) 2 − 1 = 3 + 2 − 1 = 4 Since A ( 1 ) = 4 e q 0 , A ( x ) is not divisible by ( x − 1 ) .

Evaluating B(1) Now, let's evaluate B ( 1 ) :
B ( 1 ) = 5 ( 1 ) 3 − 4 ( 1 ) 2 − 1 = 5 − 4 − 1 = 0 Since B ( 1 ) = 0 , B ( x ) is divisible by ( x − 1 ) .

Evaluating C(1) Next, let's evaluate C ( 1 ) :
C ( 1 ) = 2 ( 1 ) 3 − 3 ( 1 ) 2 + 2 ( 1 ) − 1 = 2 − 3 + 2 − 1 = 0 Since C ( 1 ) = 0 , C ( x ) is divisible by ( x − 1 ) .

Evaluating D(1) Finally, let's evaluate D ( 1 ) :
D ( 1 ) = ( 1 ) 3 + 2 ( 1 ) 2 + 3 ( 1 ) + 2 = 1 + 2 + 3 + 2 = 8 Since D ( 1 ) = 8 e q 0 , D ( x ) is not divisible by ( x − 1 ) .

Conclusion Therefore, the polynomials B ( x ) and C ( x ) are divisible by ( x − 1 ) .


Examples
Polynomial divisibility is a fundamental concept in algebra, with practical applications in various fields. For instance, in engineering, when designing filters or control systems, engineers often work with transfer functions, which are rational functions (ratios of polynomials). Knowing whether a polynomial is divisible by another can help simplify these functions, making the design and analysis process more manageable. Similarly, in cryptography, polynomial divisibility plays a role in constructing and analyzing error-correcting codes, ensuring reliable data transmission.

Answered by GinnyAnswer | 2025-07-03

The polynomials that are divisible by ( x − 1 ) are B ( x ) and C ( x ) , as both evaluate to zero when substituting x = 1 . The choices are: B , C .
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Answered by Anonymous | 2025-07-04