Answer:
10 x^3 - x^2 - 29 x - 30 = (10 x + 9)(x^2 - x - 2) + -12
Explanation:
Compute (10 x^3 - x^2 - 29 x - 30) รท (x^2 - x - 2) using polynomial long division:
x^2 - x - 2 | 10 x^3 | - | x^2 | - | 29 x | - | 30
To eliminate the leading term of the numerator, 10 x^3, multiply x^2 - x - 2 by 10 x to get 10 x^3 - 10 x^2 - 20 x. Write 10 x on top of the division bracket and subtract 10 x^3 - 10 x^2 - 20 x from 10 x^3 - x^2 - 29 x - 30 to get 9 x^2 - 9 x - 30:
| | | | | 10 x | |
x^2 - x - 2 | 10 x^3 | - | x^2 | - | 29 x | - | 30
| -(10 x^3 | - | 10 x^2 | - | 20 x) | |
| | | 9 x^2 | - | 9 x | - | 30
To eliminate the leading term of the remainder of the previous step, 9 x^2, multiply x^2 - x - 2 by 9 to get 9 x^2 - 9 x - 18. Write 9 on top of the division bracket and subtract 9 x^2 - 9 x - 18 from 9 x^2 - 9 x - 30 to get -12:
| | | | | 10 x | + | 9
x^2 - x - 2 | 10 x^3 | - | x^2 | - | 29 x | - | 30
| -(10 x^3 | - | 10 x^2 | - | 20 x) | |
| | | 9 x^2 | - | 9 x | - | 30
| | | -(9 x^2 | - | 9 x | - | 18)
| | | | | | | -12
The quotient of (10 x^3 - x^2 - 29 x - 30)/(x^2 - x - 2) is the sum of the terms on top of the division bracket. The remainder is the expression that remains after the final subtraction step.
| | | | | 10 x | + | 9 | (quotient)
x^2 - x - 2 | 10 x^3 | - | x^2 | - | 29 x | - | 30 |
| -(10 x^3 | - | 10 x^2 | - | 20 x) | | |
| | | 9 x^2 | - | 9 x | - | 30 |
| | | -(9 x^2 | - | 9 x | - | 18) |
| | | | | | | -12 | (remainder) invisible comma
(10 x^3 - x^2 - 29 x - 30)/(x^2 - x - 2) = (10 x + 9) - 12/(x^2 - x - 2)
Write the result in quotient and remainder form:
Answer: 10 x^3 - x^2 - 29 x - 30 = (10 x + 9)(x^2 - x - 2) + -12