Divide. (3x^4-2x^3+4x-5) / (x^2+4) Fill in the blanks for the quotient. (Q), and the remainder (R) separated by a comma.
In dividing polynomials, you can apply the rule in dividing whole numbers. Arrange the given in this manner (I'll just put `0x^2` in between `-2x^3` and `4x` so it will be easier to deal with later):
`x^2 + 4` / `3x^4-2x^3 + 0x^2 +4x-5`
First, ignore all other terms and just focus on the leading term of the divisor (x^2) and the leading term of the dividend (3x^4). I divide `3x^4` by `x^2` , I get `3x^2.`
Put that on top and this is how it looks like:
`x^2 + 4` /`3x^4 - 2x^3 + 0x^2 + 4x -5`
Now multiply with the divisor and put the answer underneath.
`x^2 + 4` /`3x^4 -2x^3 + 0x^2 +4x -5`
`3x^4 + 0x^3+ 12x^2`
It will be easier if you align the terms with the same coefficients. and also put zero for the missing term, like what I did on 0x^3.
Next is to subtract the answer from the multiplication from the dividend.
`x^2 +4` /`3x^4 - 2x^3 + 0x^2 +4x -5`
- `3x^4 +0x^3 +12x^2`
`0x^4 -2x^3 -12x^2 +4x -5`
Ignore the term with the zero. Then, do again the divison and put the answer on the top.
`3x^2 - 2x`
`x^2 + 4` /`3x^4 - 2x^3 +0x^2 +4x-5`
- `3x^4 +0x^3+12x^2`
`-2x^3 - 12x^2 + 4x -5`
- `-2x^3 +0x^2 -8x`
`0x^2 -12x^2 + 12x -5`
`-12x^2 +12x -5`
- `-12x^2 +0x -48`
`0x^2 +12x +43`
You can now stop there, since `12x+43` has lesser degree as the divisor. Few reminders, take note that when you subtrating you must be careful that the minus sign should be go with each term. It is just like multiplying -1 by all the terms then combine it with their similar terms on top of them.
So, the qoutient, (Q) is `3x^2 - 2x -12.`
the remainder, (R) is `12x + 43.`
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