1. ## Dividing polynomials

I'm learning how to divide polynomials by expressions like a + 1 and I'm stuck on a particular question:

4x^4 - 3x^2 + x + 2 divided by 2x + 3

The answer must be expressed in the form (2x+3)*Quotient+Remainder.

Here's how I've been going about it.

4x^4 - 3x^2 + x + 2 = (2x + 3)(ax^3 + bx + c) + R

Then after expanding and collecting like terms, I have:

2ax^4 + 3ax^3 + 2bx^2 + (2b+2c)x + 3c + R

And it's here that I get stuck. The problem here is that I don't know how to go on. What for example do I do with the x^3 bit?

Many thanks

2. ## Re: Dividing polynomials

Originally Posted by bubbletea999
I'm learning how to divide polynomials by expressions like a + 1 and I'm stuck on a particular question:

4x^4 - 3x^2 + x + 2 divided by 2x + 3

The answer must be expressed in the form (2x+3)*Quotient+Remainder.

Here's how I've been going about it.

4x^4 - 3x^2 + x + 2 = (2x + 3)(ax^3 + bx + c) + R

Then after expanding and collecting like terms, I have:

2ax^4 + 3ax^3 + 2bx^2 + (2b+2c)x + 3c + R

And it's here that I get stuck. The problem here is that I don't know how to go on. What for example do I do with the x^3 bit?

Many thanks
Have you done synthetic division? If so use it.

If not try equating coeficients of like powers in each of your expressions, to get you started (assuming you have no algebraic mistakes) for the fourth power term:

4=2a

CB

3. ## Re: Dividing polynomials

You have:
$4x^4 - 3x^2 + x + 2 = (2x + 3)(ax^3 + bx^2 + {\color{red}cx}+d) + R$

Open the brackets:

\begin{align*} 4x^4 - 3x^2 + x + 2 &=(2a)x^4+(2b)x^3+(2c)x^2+(2d)x+(3a)x^3+(3b)x^2+(3 c)x+(3d)+R \end{align*}

\begin{align*} \implies 4x^4 - 3x^2 + x + 2 &= (2a)x^4+(2b+3a)x^3+(2c+3b)x^2+(3c+2d)x+(3d+R) \end{align*}

Equating the coefficients:

$2a=4$

$2b+3a=-3$

$2c+3b=0$

$3c+2d=1$

$3d+R=2$

The correct solutions to these equations will give you the value of $a,b,c,d \ \textrm{and} \ R$.

4. ## Re: Dividing polynomials

I've now learned how to do synthetic division, and in the process of doing so also learned about the general form, which ended up solving my problem with this! I was going wrong because I wasn't accounting for the "missing power" in the original equation. It jumps from x^4 to x^2. I've solved the problem by adding 0x^3 in between them -- does this sound right? It's produced the right answer, at any rate.

Thanks again.

5. ## Re: Dividing polynomials

Another way to divide $4x^4 - 3x^2 + x + 2$ by $2x + 3$, more like "standard" numerical long division:

$2x$ divides into $4x^4$ $2x^3$ times. Multiplying that by $2x+ 3$ gives [tex](2x+3)(2x^3)= 4x^4+ 6x^3. Subtract [tex](4x^4+ 0x^3)- (4x^4+ 6x^3)= -6x^3. That leaves us $-6x^3- 3x^2+ x+ 2$. Now, $2x$ divides into $-6x^3$ $-3x^2$ times. Multiplying that by $2x+ 3$ gives $-6x^3- 9x^2$. Subtract $(-6x^3- 3x^2)- (-6x^3- 9x^2)= 6x^2$. That leaves $6x^2+ x+ 2$. $2x$ divides into $6x^2$ 3x times. Multiplying that by $2x+ 3$ gives $6x^2+ 9x$. Subtracting $(6x^2+ x)-(6x^2+ 9x)= -8x$. That leaves us $-8x+ 2$. $2x$ divides into $-8x$ -4 times. Multiplying that by $2x+ 3$ gives $-8x- 12$. Subtracting $(-8x+ 2)- (-8x- 12)= 14$.

That is, the quotient is $2x^3- 3x^2+ 3x- 4$ with remainder 14.

(Captain Black, I was under the impression that "synthetic division" could only be used to divide by something of the form x+ a.)