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Question
Sir, How do  i solve polynomial-linear word problem effectively. For example, Find the values of the constants p,q and r such that,when the polynomial f(x)=x^3+px^2+qx+r is divided by(x+2),(x-1) and (x-3),the remainders are, respectively, -48, 0 and 2. Thuse,factorize f(x) completely.

Answer
Hi Johnny,
When the polynomial f(x) is divided by x-a, the remainder is given by f(a).
For the polynomial f(x) = x + px + qx + r, and from the information provided, we have
f(-2) = (-2) + p(-2) + q(-2) + r = -48
-8 + 4p - 2q + r = -48
4p - 2q + r = -40

f(1) = (1) + p(1) + q(1) + r = 0
1 + p + q + r = 0
p + q + r = -1

f(3) = (3) + p(3) + q(3) + r = 2
27 + 9p + 3q +  r = 2
9p + 3q + r = -25

and the three equations solved simultaneously to get
p = -5, q = 8, r = -4

So,
f(x) = x - 5x + 8x - 4
As it is the case that dividing by x-1 gives a remainder of 0, we conclude that x-1 is a factor. To factorize f(x) completely, we first divide by x-1 and then factorize the resulting quadratic expression.

Now,
(x - 5x + 8x - 4) / (x-1) = x - 4x + 4
= (x-2)

Therefore,
f(x) = x - 5x + 8x - 4 = (x-1)(x-2)

Regards

Advanced Math

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