What is the Remainder When a Polynomial is Divided by a Product of Linear Terms?

[sbhit2001]

Homework Statement

If a , b, c are distinct and p(x) is a polynomial in x which leaves remainders a,b,c on division by (x-a),(x-b),(x-c) respectively. Then the remainder on division of p(x) by(x-a)(x-b)(x-c) is

Homework Equations

As it is given that p(x) gives remainder a when divided by (x-a), so p(a) should be equal to a by remainder theorem.Similarl p(b) = b and p(c)=c.

The Attempt at a Solution

As (x-a)(x-b)(x-c) is a cubic polynomial, remainder can be max quadratic so I assume it to be px^2 + qx + r.Again by remainder theorem we will get 3 equations for p,q,r by using a,b,c. As we see that p(a) = a, p(b)=b,p(c)=c ; Then we can say that a,b,c will be roots of px^2 +x(q-1) + r. But a quadratic polynomial can have max 2 roots. Can u please tell me what did I do wrong here?

 

[mfb]

Then we can say that a,b,c will be roots of px^2 +x(q-1) + r.

How did you come to that conclusion?

 

[sbhit2001]

How did you come to that conclusion?

Because (x-a)(x-b)(x-c) will be 0 at the values ie a,b,c so remainder at each of these values will be p(a),p(b),p(c) ie a,b,c. So, eg if we take a we get pa^2 + qa + r = a , ie pa^2 +a(q-1) + r = 0 . In all a,b,c we get same expression. So I think that means a , b ,c will be roots of px^2 + x(q-1) + r.

 

[Mark44]

Homework Statement

If a , b, c are distinct and p(x) is a polynomial in x which leaves remainders a,b,c on division by (x-a),(x-b),(x-c) respectively. Then the remainder on division of p(x) by(x-a)(x-b)(x-c) is

The remainder is what?

Homework Equations

As it is given that p(x) is divisible by (x-a)

This is NOT given. It does NOT say that p(x) is divisible by x - a, or x - b, or x - c.

, so p(a) should be equal to a by remainder theorem.Similarl p(b) = b and p(c)=c.

The Attempt at a Solution

As (x-a)(x-b)(x-c) is a cubic polynomial,

remainder can be max quadratic so I assume it to be px^2 + qx + r.Again by remainder theorem we will get 3 equations for p,q,r by using a,b,c. As we see that p(a) = a, p(b)=b,p(c)=c ; Then we can say that a,b,c will be roots of px^2 +x(q-1) + r. But a quadratic polynomial can have max 2 roots. Can u please tell me what did I do wrong here?

 

[ack]

Id like you to consider this eg: let p(x)=x and when you divide this by x-a,x-b,x-c you get a,b,c as remainders and when you divide this by their combined product you get x ie p(x) itself as the remainder.So I think your remainder will be p(x) in your question ie the degree of p(x) is < 3.