# Factors of a polynomial

$$3x^2 + 4x + C \equiv A(x + 1)^2 + B(x + 1) + 7$$
Find all values of A, B and C.
Could someone teach how to do this?

## Answers and Replies

arildno
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Rewrite
$$x^{2}=((x+1)-1)^{2}$$
and do the necessary operations.

Huh? Sorry I don't understand what u mean.

arildno
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$$((x+1)-1)^{2}=(x+1)^{2}-2(x+1)+1$$

How is $$((x+1)-1)^{2}=(x+1)^{2}-2(x+1)+1$$
related to $$3x^2 + 4x + C \equiv A(x + 1)^2 + B(x + 1) + 7 ?$$
Sorry but i don't get u.

arildno
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It's equal to $$x^{2}$$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Make a similar rewriting of x:
$$x=((x+1)-1)$$
Now, substitute these expressions for $$x,x^{2}$$ into your LEFT-HAND SIDE.
Reorganize the terms you get, and derive conditions so that your new expression equals your ORIGINAL RIGHT-HAND SIDE.
This will determine A,B,C.

Hurkyl
Staff Emeritus
Gold Member
Alternatively, you could expand the right hand side, and equate coefficients to solve for A, B, and C.
Or you could plug in a few values for x to generate equations.

I probably wasn't clear. Firstly, how does $$x^{2}=((x+1)-1)^{2}.$$ Secondly, which ones do i substitute in for x? Is it $$((x+1)-1)^{2} = 3 or (x + 1)^2 ?$$

arildno
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You know that 1-1=0, right?
So (x+1)-1=x+1-1=x.
To be nice, I'll do this for once:
We have:
$$3x^{2}=3((x+1)-1)^{2}=3(x+1)^{2}-6(x+1)+3$$
$$4x=4((x+1)-1)=4(x+1)-4$$
$$C=C=C$$
Now, add these equations together, downwards. The outermost terms then turn into:
$$3x^{2}+4x+C=3(x+1)^{2}-2(x+1)+(C-1)$$
Do you understand this?

Hurkyl
Staff Emeritus
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What does (x + 1) - 1 equal? Then what does ((x+1) - 1)^2 equal?

I'm just curious, has this got something to do with modulus?

arildno
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Sariaht said:
I'm just curious, has this got something to do with modulus?
Not that I know of..
It is simply to substitute "equal for equal"

<Sarcastic mode ON>
arildno you have to be a little more diplomatic, I think
<Sarcastic mode OFF>

you guys are making the problem too complicated
since
$$3x^2 + 4x + C = A(x + 1)^2 + B(x + 1) + 7$$
expand $$A(x + 1)^2 + B(x + 1) + 7$$
then you have $$Ax^2 + 2Ax + A + Bx + B + 7$$
and $$3x^2 + 4x + C = Ax^2 + 2Ax + A + Bx + B + 7$$
equate the coeffients of the powers you have
$$3 = A$$ for $$x^2$$
$$2A + B = 4$$ for $$x^1$$
$$A + B + 7 = C$$ for $$x^0$$
solve the system and you got the answer

Last edited:
Ahh... Sorry but i was a bit slow on "(x+1)-1" part (sorry if i pissed u arildno ). Thanks for the help guys!

arildno