Simplyfying (Indentitied related)

[ASMATHSHELPME]

Simple for you guys i guess, But tough for me - Guess I am just thick!

$x^4 +Ax^3 + 5x^2 + x + 3 = (x^2 +4)(X^2 -x +B) +Cx +D$

I get:

$x^4 +Ax^3 + 5x^2 + x + 3 = X^4 -x^3 - 4x^2 + Bx^2 - 4x + 4B + Cx + D$

Now, I think i need to simplify this more because i can't compare co-efficients can i?

Can someone run me through the further simplifications?

Maybe $Bx^2 + 4x^2$ into $(4+B)X^2$ ? Is this wise and possible? What else?

Need to learn simplification better, Finding my basic maths is poor so Alevel is tough!

 

[hypermorphism]

The law of distribution of multiplication over addition: a*(b + c) = a*b + a*c. The equality sign means that any expression of the form of the right hand side may be replaced by the expression on the left hand side (and vice versa) and still maintain the truth of the original expression. As long as X=x in your expression, what you're doing is fine.

 

[iNCREDiBLE]

You're on the right path, my friend!

$$x^4 +Ax^3 + 5x^2 + x + 3 = (x^2 +4)(x^2 -x +B) +Cx +D$$
<=>
$$x^4 +Ax^3 + 5x^2 + x + 3 = x^4 - x^3 + Bx^2 + 4x^2 - 4x + 4B +Cx +D$$
<=(cancellation & simplification)=>
$$Ax^3 + 5x^2 + x + 3 = - x^3 + (B+4)x^2 + (C-4)x + 4B + D$$

 

[HallsofIvy]

Now recall that if that is true for all x, then the corresponding coefficients must be equal. You can just look at that and see what A must be!