Question

One of the factors of (a-3)²+5(3-a) is

Answer 1

Answer:

Standard Solution:

Let a4+2a3+3a2+2a+1=(ba2+ca+d)2 where b,c,d∈R

Observe that

(ba2+ca+d)2=b2a4+2bca3+(2bd+c2)a2+2cda+d2

Comparing coefficient of like terms, we get

b2=1⇒b=±1

bc=1⇒c=±1

±2d+1=3⇒d=±1

Thus,

a4+2a3+3a2+2a+1=[±(a2+a+1)]2

Note: Although it appears that this method assumes that the quartic must be the square of a polynomials, but if you end up getting no answer, it means you arrived at a contradiction and the equation is not the square of a polynomial. Moreover, the if there are 2 answers, they are additive inverses of each other.