Let P(x)=(x^ 2 +px+q)(x-2)+r

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Let

P(x)=(x^{2}+px+q)(x-2)+r,

where

p

q

, and

r

are real numbers. (i) Expand

P(x)

and collect like terms in descending powers of

x

. Express every coefficient of the polynomial in terms of

p

q

, and

r

. (ii) The coefficient of

x^{2}

P(x)

is required to be

5

, and the constant term is

-3

. Write the system of equations that these conditions place on

p

q

, and

r

. (iii) Solve the system obtained in part (ii) to find all possible triples

(p,q,r)

satisfying the given conditions. (iv) For each triple found in part (iii), write the complete polynomial

P(x)

and state its leading coefficient.