Question

Help me please :)

1. 2x^4+5x^3+x-1 by x^2-2x+1

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

Solve using polynomial long division and show work

Answer 1

Do you know how to do polynomial long division?

Answer 2

No

Answer 3

Would you like to learn how to do it?

Answer 4

If you wanna teach me

Answer 5

Alright, let's start with \((2x^4+5x^3+x-1) / (x^2-2x+1) \).

Answer 6

We start with the first terms. \(2x^4/x^2 = ?\)

Answer 7

Could you do this division?

Answer 8

So \(2x^4/x^2 = 2x^2\).

Answer 9

Okay

Answer 10

Can you multiply polynomials? Now we multiply \(2x^2\times (x^2-2x+1)\).

Answer 11

\[
2x^2\times(x^2-2x+1) = 2x^4-4x^3+2x^2
\]

Answer 12

We subtract this from the thing being divided.
\[
2x^4+5x^3+x-1 - (2x^4-4x^3+2x^2) = 9x^3-2x^2+x-1
\]

Answer 13

Basically what I've done so far is the first step.

Answer 14

Ooo okay

Answer 15

Since the divisor starts with \(x^2\), and the remaining part starts with \(9x^3\), we can still keep going. Now we do \(9x^3/x^2 = 9x\).

Answer 16

\(9x(x^2-2x+1) = 9x^3 -18x^2+9x\)

Answer 17

\[
9x^3-2x^2+x-1 - ( 9x^3 -18x^2+9x) = 16x^2-8x-1
\]

Answer 18

It keeps going like this. Does this make sense?

Answer 19

We can still keep going since the starting parts are \(x^2\) and \(16x^2\).

Answer 20

\(16x^2/x^2=16\).

\(16(x^2-2x+1) = 16x^2-32x+16\).

\(16x^2-8x+1 - (16x^2-32x+16) = 24x-15 \).

Answer 21

At this point we have to stop, because the starting parts are \(x^2\) and \(24x\). Since the divisor has a larger exponent, we simply accept it is a remainder. Thus we end with \[
\frac{24x-15}{x^2-2x+1}
\]. If we add this to the parts we already divided out earlier, the final answer is: \[
2x^2+9x+16+\frac{24x-15}{x^2-2x+1}
\]

Answer 22

The algorithm here is:
Step 1: Take the dividend's largest term and divide by the divisors largest term.
Step 2: Multiply Step 1's result with the divisor.
Step 3: Subtract Step 2's result from the dividend.
Continue until the dividend's largest term is lower than the divisors largest term.

Answer 23

I think it would be great if you tried to do question 2 on your own, but I can help if you still don't get it.

Answer 24

Okay thank you so much <33

Answer 25

https://www.mathsisfun.com/algebra/polynomials-division-long.html
This website also explains it a bit.

Answer 26

Thank you bb