How Do You Solve Polynomial Equations Using the Factor and Remainder Theorem?

[DizzyDoo]

A couple of hard questions about the Factor and Remainder Thoerem that I'm having a hard time with.

Homework Statement

18) f(x) = 2x3 + x2 – 5x + c, where c is a constant.
Given that f(1) = 0,

(a) find the value of c

(b) factorise f(x) completely,

(c) find the remainder when f(x) is divided by (2x – 3).

19) f(x) = x3 – 2x2 + ax + b, where a and b are constants.

When f(x) is divided by (x – 2), the remainder is 1.

When f(x) is divided by (x + 1), the remainder is 28.

(a) Find the value of a and the value of b.

(b) Show that (x – 3) is a factor of f(x).

Homework Equations

None

The Attempt at a Solution

These are the last two questions of the homework, and the only ones I am having some difficulty with. The other 17 questions I have finished and am happy with. I reckon I can do 18) a) though;

f(1) = 2(1)^3 + 1² - 5(1) + c
= 2 + 1 -5 + c
c = 2

Right? Any help with the other questions is very welcome! Thanks for your time.

 

[rock.freak667]

Yes c=2 making $f(x)=2x^3 + x^2-5x+2$

The remainder and factor theorem states that if f(x) is any polynomial and f(x) is divided by x-a then the remainder is f(a).If f(a)=0 then (x-a) is a factor of f(x).

From the theorem above.

f(1)=0. This means that (x-1) is a factor of f(x). Now you can just do long division or synthetic division and you will get the other quadratic factor. which you can then factorize further if possible.

 

[DizzyDoo]

Hey thanks, that's a great explanation, I've finished off question 18 quite comfortably. Any help with question 19?

 

[Mathdope]

19) f(x) = x3 – 2x2 + ax + b, where a and b are constants.

When f(x) is divided by (x – 2), the remainder is 1.

When f(x) is divided by (x + 1), the remainder is 28.

They are leading you to a point where you will have two equations in two unknowns.

By the remainder theorem, the first part says that f(x) = (x-2)q(x) + f(2), where q(x) is a quadratic polynomial. This tells you something about f(2). The second part says something similar. Can you put the rest together?

(a) Find the value of a and the value of b.

This should be accomplished using the two facts above in combination with the remainder theorem.

(b) Show that (x – 3) is a factor of f(x).

Once you know a and b you should be able to prove this - remember what it means about f(3).