Question

When a polynomial f(x) is divided by x-3 and x+6 , the respective remainders are 7 and 22. What is the remainder when f(x) is divided by (x-3)(x+6)
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Answer 1

Answer:

When a polynomial f(x) is divided by x-3 and x+6, the respective remainders are 7 and 22. What is the remainder when f(x) is divided by (x-3) (x+6)?

Let’s look at a more general problem.

We know that the remainders when f(x) is divided by x−a and x−b are r and s respectively. What’s the remainder when f(x) is divided by (x−a)(x−b) ?

Let’s assume first that a≠b and write px+q the sought remainder. This means that, for some polynomial g(x) ,

f(x)=(x−a)(x−b)g(x)+px+q(*)

The hypotheses are equivalent to f(a)=r and f(b)=s , so we can evaluate (*) at a and b respectively, getting

{pa+q=rpb+q=s

This solves easily, by subtracting the second equation from the first, so p(a−b)=r−s and

p=r−sa−b

Next

q=r−pa=r−r−sa−ba=as−bra−b

Hence the remainder is

r−sa−bx+as−bra−b

You can now substitute a=3 , b=−6 , r=7 and s=22 , getting

−53x+12

Of course, if a=b the problem is underdetermined.

When ever a polynomial of degree N is divided by another polynomial of degree < N, the remainder will always be a polynomial ONE degree less than degree of denominator.

Remainder Theorem states that if a function f(x) is divided by (x-a), then f(a) is the remainder.

Taking cognizance of above two facts, we know the remainder when f(x) is divided by (x-3)(x+6) will be linear polynomial of degree ONE.

Let the remainder be represented by Ax + B

If f(x) is divided by x-3, remainder is 7

=> 3A + B = 7

If f(x) is divided by x-(-6), remainder is 22

=> -6A + B = 22

Solving the two equations, we get A = -15/9 & B = 12

So final remainder is -15x/9 + 12

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Answer 2

Answer:

f(x) = (x – 3)(some polynomial) + 7 so that f(3) = 7

f(x) = (x + 6)(some other polynomial) + 22 so that f( – 6 ) = 22

when f(x) is divided by a quadratic factor the remainder is of the form ax + b

f(x) = (x – 3)(x + 6)(yet another polynomial) + ax + b

subs x = 3: 7 = 3a + b

subs x = – 6: 22 = – 6a + b

subtracting these we get – 15 = 9a