Synthetic Division for Higher Order Polynomials

[zzmanzz]

Homework Statement

So I thought I knew how to do synthetic division but ran into this problem

4a^4+4a^3-9a^2-4a+16 / (a^2-2)

Homework Equations

The Attempt at a Solution

[/B]
All the examples I can find don't have a second degree polynomial in the denominator. i.e. they are a-3 or a+2. How do you go about doing this division with higher order polynomials in the denominator?

Like do I still set it up as ?

-2 | 4 4 -9 -4 16
Thanks

 

[fresh_42]

Here's an example I recently wrote here:
https://www.physicsforums.com/threa...r-a-polynomial-over-z-z3.889140/#post-5595083

It is basically the exact same thing as with numbers. As a hint: as long as it's new to you, proceed step by step and watch out not to mess up the signs.

Edit: Of course your example starts with $4a^4 : (a^2-2) = 4a^2 + ...$ and then the subtraction of $4a^2 \cdot (a^2-2)= 4a^4-8a^2$

 

[Mark44]

Homework Statement

So I thought I knew how to do synthetic division but ran into this problem

4a^4+4a^3-9a^2-4a+16 / (a^2-2)

Homework Equations

The Attempt at a Solution

[/B]
All the examples I can find don't have a second degree polynomial in the denominator. i.e. they are a-3 or a+2. How do you go about doing this division with higher order polynomials in the denominator?

Like do I still set it up as ?

-2 | 4 4 -9 -4 16
Thanks

As far as I know, synthetic division can be performed only when the divisor is a first degree polynomial whose leading coefficient is 1. IOW, the divisor has to be x - a, with a being either positive or negative.

The work that @fresh_42 showed is polynomial long division. Synthetic division is a special case of polynomial long division.

 

[SammyS]

Homework Statement

So I thought I knew how to do synthetic division but ran into this problem

$\ 4a^4+4a^3-9a^2-4a+16 / (a^2-2)$

Homework Equations

The Attempt at a Solution

[/B]
All the examples I can find don't have a second degree polynomial in the denominator. i.e. they are a-3 or a+2. How do you go about doing this division with higher order polynomials in the denominator?

Like do I still set it up as ?

-2 | 4 4 -9 -4 16
Thanks

I suppose you mean
$\ (4a^4+4a^3-9a^2-4a+16) / (a^2-2)$
If that were division by $\ a - 2\,,\$ then you would have positive 2 out front in your set-up for synthetic division.

But yes, you can do some form of synthetic division here. The divisor only has terms of even degree, so it interacts with the even degree terms independently from the odd degree terms.

You can look at this as $\displaystyle \ \frac{4a^4-9a^2+16}{a^2-2} + a\frac{4a^2-4}{a^2-2} \,.$

Do two individual synthetic divisions.

 

[lurflurf]

$$

\begin{array}{rr|rrrrr}
& & 4&4 & -9 & -4 & 16 \\
\hline
& 2 & & & 8 & *&*&%8 &-2
\\
0 & & & 0&*&*&%0 & 0 &
\\
\hline

&&4&*&*&*&*&%{4}&-1& 4&14
\end{array}

$$

The numbers to the left multiply the bottom left numbers to yield the middle numbers
top and middle numbers add to give bottom numbers
try to fill in the rest