Solve using long polynomial division
(5x^2-x^2+x-2)/(x^2+2)
the first step is to fix your typo. Then,
5x^3 = x^2 * 5x, so the first term in the quotient is 5x.
(5x^3 - x^2 + x - 2) - 5x(x^2+2) = -x^2-9x-2
so, the next term will be -1
can you finish it up?
google "polynomial division calculator" to find several handy online sites to confirm your work
oh i meant to say (5x^4-x^2+x-2)/(x^2+2)
oh, well. The steps remain the same.
To solve the given problem using long polynomial division, follow these steps:
Step 1: Arrange the given polynomial in descending order of exponents.
Given polynomial: (5x^2 - x^2 + x - 2)/(x^2 + 2)
Step 2: Divide the term with the highest exponent from the numerator by the term with the highest exponent from the denominator.
5x^2 divided by x^2 gives 5.
So, the first term of the quotient is 5.
Step 3: Multiply the entire denominator by the first term of the quotient and subtract it from the numerator.
The result after subtracting is:
(5x^2 - x^2 + x - 2) - (5 * (x^2 + 2))
This simplifies to:
(5x^2 - x^2 + x - 2) - (5x^2 + 10)
Simplifying further:
5x^2 - x^2 + x - 2 - 5x^2 - 10
Now, combine like terms:
(5x^2 - 5x^2) - x^2 + x - 2 - 10
This results in:
- x^2 + x - 12
Step 4: Determine the next term of the quotient.
Now, considering the numerator as - x^2 + x - 12, divide the term with the highest exponent (-x^2) by the term with the highest exponent (x^2) from the denominator.
(-x^2) divided by (x^2) gives -1.
So, the next term of the quotient is -1.
Step 5: Multiply the entire denominator by the next term of the quotient (-1) and subtract it from the numerator (-x^2 + x - 12).
The result after subtracting is:
(-x^2 + x - 12) - ((-1) * (x^2 + 2))
This simplifies to:
(-x^2 + x - 12) - (-x^2 - 2)
Simplifying further:
-x^2 + x - 12 + x^2 + 2
Now, combine like terms:
(-x^2 + x^2) + x - 12 + 2
This results in:
x - 10
Step 6: Determine the last term of the quotient.
Considering the numerator (x - 10), divide the term with the highest exponent (x) by the term with the highest exponent (x) from the denominator.
(x) divided by (x) gives 1.
So, the last term of the quotient is 1.
Step 7: Multiply the entire denominator by the last term of the quotient (1) and subtract it from the numerator (x - 10).
The result after subtracting is:
(x - 10) - (1 * (x^2 + 2))
This simplifies to:
(x - 10) - (x^2 + 2)
Simplifying further:
x - 10 - x^2 - 2
Now, combine like terms:
(x - x) - 10 - 2 - x^2
This results in:
-12 - x^2
Step 8: Finalize the quotient.
Since -12 - x^2 has no term with a higher exponent than the denominator (x^2 + 2), we cannot divide any further.
Hence, the final quotient is:
5 - 1 + (x - 10)/(x^2 + 2) = 5 - 1 - (x^2 + 12)/(x^2 + 2)