The correct conclusion is the remainder of p(x) divided by (x-1) is 5 and this can be determined by using the factorization method.
Given :
- [tex]\rm p(x)=-2x^3+3x+4[/tex]
The following steps can be used in order to determine which conclusion is correct:
Step 1 - The given expression (p(1) = 5) implies that at (x = 1) the value of p is 5.
Step 2 - According to the conclusion (A) the remainder of p(x) divided by (x-1) is 5 that is:
[tex]\rm p(x) = \dfrac{-2x^3+3x+4-5}{x-1}[/tex]
Simplify the above expression.
[tex]\rm p(x) = \dfrac{-2x^3+3x-1}{x-1}[/tex]
Now, factorize the numerator.
[tex]\rm p(x) = \dfrac{-2x^3+2x^2-2x^2+2x + x - 1}{x-1}[/tex]
[tex]\rm p(x) = \dfrac{(x-1)(-2x^2-2x+1)}{(x-1)}[/tex]
So, this conclusion is true.
Step 3 - According to the conclusion (B), the remainder of p(x) divided by (x-5) is 1 that is:
[tex]\rm p(x) = \dfrac{-2x^3+3x+4-1}{x-1}[/tex]
The numerator does not factorize. Therefore, this conclusion is false.
For more information, refer to the link given below:
https://brainly.com/question/17822016
