Find A and B for $A(x^{2}+3)+B(x^{3}+2)=5x^{4}+9x^{2}+4x$

Question

Given:$$A(x^{2}+3)+B(x^{3}+2)=5x^{4}+9x^{2}+4x$$

How does one find A and B ?

The answer is: $$A = 3x^{2};B=2x$$ but I can't see how one solves this. I tried subbing in values of x but it didn't lead anywhere.

Answer 1

Hint:

Since the LHS is of degree $4$, we can take $A=ax^2+bx+c$ and $B=mx+n$.

Now write the given identity and equate the coefficients of the same powers of $x$. You find a linear system of $5$ equations in the $5$ unknowns $a,b,c,m,n$.

Answer 2

One way to show how you can get this result:

$$5x^4+9x^2+4x=3x^4+9x^2+2x^4+4x=3x^2(x^2+3)+2x(x^3+2)$$

Compare both expressions to get A and B.

Answer 3

Polynomial division by $x^2+3$ gives $$B(-3x+2)\equiv 4x+18\pmod {x^2+3}$$ Multiplying with $(+3x+2)$ gives $$31B\equiv 62x\pmod {x^2+3}$$ so that we conclude $$ B\equiv 2x\pmod{x^2+3}.$$ Plugging in $B=2x$ (or if you want $B=2x+(x^2+3)C$ for any polynomial $C$) allows you to sove for $A$ (i.e., will produce a multiple of $x^2+3$).