Divide.
(9x^3-10+19+6x^4)/(-2x^2+3x-3)
Write your answer in the form Q(x)+ R(x)/ -2x^2+3x-3,
where Q(x) is the quotient and R(x) is the remainder.
What in particular do you need help with?
Use long division as you would with any
other long division problem.
If you post your answer I can check it for you.
To divide the given expression (9x^3-10+19+6x^4) by (-2x^2+3x-3), we can use polynomial long division. Let's break down the steps:
Step 1: Start by dividing the first term of the numerator (6x^4) by the first term of the denominator (-2x^2). The quotient is 6x^2.
Step 2: Multiply the entire denominator (-2x^2+3x-3) by the quotient we just obtained (6x^2) and subtract it from the numerator (9x^3-10+19+6x^4 - 6x^2(-2x^2+3x-3)). Simplifying this gives us:
9x^3 - 10 + 19 + 6x^4 - 6x^2(-2x^2+3x-3) = 9x^3 - 10 + 19 + 6x^4 + 12x^4 - 18x^3 + 18x^2
Step 3: Now, bring down the next term in the numerator into the expression obtained from the previous step. Here, the next term is -18x^3, and we add it to what we already have:
9x^3 - 10 + 19 + 6x^4 + 12x^4 - 18x^3 + 18x^2 + (-18x^3) = 9x^3 - 10 + 19 + 6x^4 + 12x^4 - 18x^3 - 18x^3 + 18x^2
Step 4: Repeat steps 1-3 until we finish dividing all the terms.
Continuing in a similar manner, we divide (-18x^3) by (-2x^2), resulting in 9x. We then multiply (-2x^2+3x-3) by 9x and subtract it from the previous expression:
9x^3 - 10 + 19 + 6x^4 + 12x^4 - 18x^3 - 18x^3 + 18x^2 - (9x)(-2x^2+3x-3)
Simplifying this further yields:
9x^3 - 10 + 19 + 6x^4 + 12x^4 - 18x^3 - 18x^3 + 18x^2 - (-18x^3 + 27x^2 - 27x)
Step 5: Bring down the next term from the numerator, which is -27x:
9x^3 - 10 + 19 + 6x^4 + 12x^4 - 18x^3 - 18x^3 + 18x^2 - (-18x^3 + 27x^2 - 27x) - 27x
Simplifying this gives:
9x^3 - 10 + 19 + 6x^4 + 12x^4 - 18x^3 - 18x^3 + 18x^2 - (-18x^3 + 27x^2 - 27x) - 27x = 12x^4 + 9x^3 - 37x + 46x^2
Step 6: We have now completed the division, and the expression we obtained is the quotient. The remainder is zero (there are no terms left to bring down). Therefore, the final answer can be written as:
Q(x) = 12x^4 + 9x^3 - 37x + 46x^2
R(x) = 0 (remainder)
Dividend = Q(x) + R(x) = 12x^4 + 9x^3 - 37x + 46x^2 + 0
Divisor = -2x^2 + 3x - 3
Therefore, the answer is Q(x) = 12x^4 + 9x^3 - 37x + 46x^2 and R(x) = 0, with the divisor -2x^2 + 3x - 3.