Question

findp(a-3) if p(x)=3X+2x^2-x^3

Answer 1

Factor p(x)=3x+2x²-x³=x(3+2x-x²)=x(3-x)(1+x). Now replace x with a-3:

p(a-3)=(a-3)(3-a+3)(1+a-3)=(a-3)(6-a)(a-2)=

(a-3)(6a-12-a²+2a)=(a-3)(8a-12-a²)=a(8a-12-a²)-3(8a-12-a²)=

8a²-12a-a³-24a+36+3a²=-a³+11a²-36a+36.

So p(a-3)=-a³+11a²-36a+36.

QUICK CHECK: When x=0, p(0)=0, so when a-3=0, that is, a=3 we should get 0 when we plug a=3 into p(a-3):

p(3-3)=p(0)=-27+99-108+36=135-135=0✔️ This proves the answer (in bold).

NOTE: You don't have to factor p(x); you can simply replace x with a-3 in 3x+2x²-x³, but you would end up with more terms to add or subtract and you could easily make a mistake.