Consider the quadratic polynomial where are integers and .

MATHEMATICS · Multi-part · AoPS · Mathematics · Quadratic Equations - Part 1 · Quadratic Equations - Part 1

Problem

Consider the quadratic polynomial

ax^{2}+bx+c

where

a, b, c

are integers and

a\neq 0

. (i) State the relationships that two integers

m

and

n

must satisfy so that the polynomial can be expressed as

ax^{2}+bx+c=ax^{2}+mx+nx+c

and then factored into two linear factors. (ii) Using the relationships obtained in part (i), factor the quadratic polynomial

6x^{2}+13x+6

completely into linear factors. (iii) Expand your answer to part (ii) and verify that it is identical to the original polynomial.