Question

divide f(x) by d(x) and write a summary statement in the form indicated
f(x)=x^4+5^3-2x^2+5x-3
d(x)=x^2+1

Answer 1

shouldn't that be \[f(x) = x^4+5x^3-2x^2+5x-3\]?

Answer 2

what is the first term of \(f(x)\) divided by the first term of \(d(x)\)? That will be the first term of your answer. Then multiply that term by \(d(x)\) and subtract the result from \(f(x)\). Now repeat the process with the new polynomial.

Answer 3

yes, that's right, my bad. and i don't understand... @whpalmer4

Answer 4

Okay. it's just like doing long division on numbers, except maybe easier!

So, what is the first term of \(f(x)\)?

What is the first term of \(d(x)\)?

Answer 5

x^2 and x^2

Answer 6

one of those is correct, but the other is not.

\[f(x) = {\huge x^4} + 5x^3-2x^2+5x-3\]\[d(x) = {\huge x^2}+1\]

Answer 7

yes! my bad!

Answer 8

you just divide the first term? which means subtract the powers? x^2?

Answer 9

@whpalmer4

Answer 10

Yes, \(x^4/x^2 = x^{4-2} = x^2\)

So the first term of our answer is \(x^2\)

Now we multiply that term by \(x^2+1\). \[x^2(x^2+1) = \]

Answer 11

\[x^4 +x^2?\]

Answer 12

Yes. Now we subtract that from \(f(x)\) to see what we have left.

f(x) - (x^4+x^2) = x^4+5x^3-2x^2+5x-3 - (x^4+x^2) =

Answer 13

\[f(x) - (x^4+x^2) = x^4+5x^3-2x^2+5x-3 - (x^4+x^2) =\]

Answer 14

not my best day! need more coffee :-)

Answer 15

\[5x^3-x^2+5x+3?\]

Answer 16

and it's no problem!

Answer 17

:)

Answer 18

Hmm, close, but no cigar...

What happened to the x^2 term?

Answer 19

we're subtracting \((x^4+x^2)\) so that minus sign also is applied to the \(x^2\):

\[x^4+5x^3-2x^2+5x-3-(x^4+x^2) = x^4+5x^3-2x^2+5x-3-x^4-x^2\]\[=x^4-x^4+5x^3-2x^2-x^2+5x-3 = 5x^3-3x^2+5x-3\]

For this very reason, I prefer to multiply the result by -1 and add instead of trying to subtract.

\[-1*x^2(x^2+1) = -x^4 - x^2\]
\[f(x) + (-x^4 - x^2) = x^4+5x^3-2x^2+5x-3-x^4-x^2\]\[=5x^3-3x^2+5x-3\]

Answer 20

Now we repeat the process with our new, smaller polynomial. What is the first term divided by the first term of \(d(x)\)?

Answer 21

x^2

Answer 22

\[5x^3/x^2 = x^2\]?

Answer 23

5x!

Answer 24

Good. The next term of our answer is \(5x\). What is \(5x*d(x)\)? That's what we'll subtract from \(5x^3-3x^2+5x-3\)

Answer 25

5x^3+5x^2

Answer 26

mmm....no. try again :-) \[5x*d(x) = 5x*(x^2+1) = \]

Answer 27

5x^3+5x!

Answer 28

Right. and when we remove that from what we have left of \(f(x)\) what do we get?

Answer 29

would you be subtracting that from f(x)?

Answer 30

no, from \(5x^3-3x^2+5x-3\)

Answer 31

so -3x^2-3?

Answer 32

Yes. Can you do the next step? Just like the first two...

Answer 33

what is the first term of \(-3x^2-3\) divided by the first term of \(d(x)\)?