Find f(x) if f(x)=2x²+4x-f(x-3)

Question

Find f(x) as a polynomial in x only. (HINT: f(x) must logically be a polynomial of the same degree.)

Comments

Oh boy......I forgot how to do this. 

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What did you forget how to do?

Answer 1

Let f(x)=ax²+bx+c,

then f(x-3)=a(x-3)²+b(x-3)+c=ax²-6ax+9a+bx-3b+c.

f(x)+f(x-3)=ax²+bx+c+ax²-6ax+9a+bx-3b+c=2x²+4x,

Therefore, 2ax²+x(2b-6a)+(2c+9a)=2x²+4x.

Matching coefficients:

x²: 2a=2, so a=1;

x: 2b-6a=4, 2b-6=4, 2b=10, b=5;

constant: 2c+9a=0, 2c+9=0, c=-9/2.

f(x)=x²+5x-9/2.

Check it out: f(x-3)=x²-6x+9+5x-15-9/2=x²-x+9/2.

Add x²+5x-9/2: 2x²+4x.

So f(x-3)+f(x)=2x²+4x; therefore f(x)=2x²+4x-f(x-3).