Understanding the Remainder Theorem for Polynomial Division

[Jen23]

Homework Statement

What is the remainder when -3x^3 + 5x - 2 is divided by x?

The Attempt at a Solution

Not sure how to complete this one, I would assume that it is the same as x+0? How would you divide the last term, (-2). Please show your steps as this will help me a lot! Thanks!

 

[SammyS]

Homework Statement

What is the remainder when -3x^3 + 5x - 2 is divided by x?

The Attempt at a Solution

Not sure how to complete this one, I would assume that it is the same as x+0? How would you divide the last term, (-2). Please show your steps as this will help me a lot! Thanks!

Please show us your steps. Then we can comment on the situation.

Is it bothering you that a remainder is negative? (b/c, not allowed for integer division.)

 

[symbolipoint]

Question seems strange. Usually you want to divide by a binomial or some other polynomial. Here your divisor is to be just a monomial.

(-3x^3+5x-2)/x

-3x^2+5-2/x

Remainder, if that's what it is, looks like -2/x.

See that your given polynomial dividend does not show a factor of x.

 

[Jen23]

Please show us your steps. Then we can comment on the situation.

Is it bothering you that a remainder is negative? (b/c, not allowed for integer division.)

-3x^3 / x = -3x^2
0x^2 / x = +0x
5x / x = 5
-2 / x = 0 <---- (I think I solved this one incorrectly but not quiet sure. Would the answer just be -2/x sine we cannot divide it?)

Therefore the quotient that I get in the end is -3x^2 + 5. My remainder ended up being -2.
Sorry if my format is incorrect, it's difficult to type out long division on here.
I know that it is a negative, and I double checked to see if I had written the question correctly, which I did. This is a practice question from the book and I just needed help solving it.

 

[SammyS]

-3x^3 / x = -3x^2
0x^2 / x = +0x
5x / x = 5
-2 / x = 0 <---- (I think I solved this one incorrectly but not quiet sure. Would the answer just be -2/x sine we cannot divide it?)

Therefore the quotient that I get in the end is -3x^2 + 5. My remainder ended up being -2.
Sorry if my format is incorrect, it's difficult to type out long division on here.
I know that it is a negative, and I double checked to see if I had written the question correctly, which I did. This is a practice question from the book and I just needed help solving it.

Your quotient and remainder are correct.

 

[FactChecker]

-2 / x = 0 <---- (I think I solved this one incorrectly but not quiet sure. Would the answer just be -2/x sine we cannot divide it?)

Therefore the quotient that I get in the end is -3x^2 + 5. My remainder ended up being -2.

The "remainder" terminology is a little difficult. I don't know if you should say the remainder is -2 or -2/x. It's one or the other. I suspect that -2 would be better to call a remainder, since "remainder" is usually not divided by the divisor when integer long division is done.

 

[LCKurtz]

If you take a polynomial $P(x)$ and divide it by polynomial divisor $D(x)$ to get a Quotient $Q(x)$ and a remainder $R(x)$ the result of your division can be written$$
P(x) = Q(x)D(x) + R(x)$$In our case$$
-3x^3+5x-2 = (-3x+5)x -2$$and the remainder is $-2$, not $\frac {-2} x$.

 

[symbolipoint]

To make the sense clearer, try any simple example of regular whole number division having a remainder. Remind yourself of how the remainder is related to the divisor. Further my trying to discuss this would be messy but still you'll get the right understanding without my trying to. You can also re-read what your College Algebra book shows you about Rational Roots theorem and how it discusses your polynomial, divisor, roots, and remainders.