Question

I just need hints

Answer 1

Let f(x) be a polynomial function. If f(x) is divided by x-1 , x+1 and x+2, then remainders are 5,3 and 2 respectively. When f(x) is divided by
x^3 +2x^2 -x -2 , then the remainder is ?

Answer 2

@amistre64

Answer 3

I know Factor theorem and remainder theorem

Answer 4

Hint: Multiply the given factors, (x-1), (x+1) and (x+2), together. What polynomial do you obtain from having done so? Does knowing the answer to that question help at all?

Have you considered what order the polynomial f(x) may have?

Have you considered writing a model for this polynomial in the form y=ax^2+bx+c, or z=ax^3+bx^2+cx+d and using synthetic division and the information about the remainders to actually obtain a, b and c (or a, b, c and d)?

Answer 5

x^3+2x^2-x-2

Answer 6

Note: I have not yet solved this problem myself, but do believe that the following may be helpful to you in finding your own solution.

Just supposing that our polynomial were y=ax^2+bx+c (which has three unknown coefficients, a, b and c).

Supposing we were to divide this poly by the first given factor, x-1, using synth. div. We'd obtain:

1 | a b c
a a+b
------------------
a a+b a+b+c

and from the info given in the problem statement, this first remainder is equal to 5.

Do the same thing for the next two divisors (x+1) and (x+2).

You will end up with three equations in the three unknowns, which should be solvable.

Knowing a, b and c, you'll be able to write out f(x).

Then try dividing f(x) by the given divisor x^3+2x^2-x-2. What is the value of the remainder, if any?

Answer 7

dividing x-1 i got 0 as the remainder

Answer 8

Let \(f(x)\) be a polynomial

then based on the information above it has the form

\[f(x)=a(x)(x-1)(x+1)+b(x)(x-1)(x+2)\]
\[+c(x)(x+1)(x+2)+d(x)(x-1)(x+1)(x+2)\]

then \[f(1)=c(1)\cdot(1+1)(1+2)=5\]

\[f(-1)=b(-1)\cdot(-1-1)(-1+2)=3\]


ect...


doing it this way you can get the answer (which I have) for any polynomial that satisfies the above conditions.