Divide (x^6-12x^3-35)/(x^3-7)
Make u = x^3
Then you have
(u^2 - 12u - 35)/(u-7)
You sure it's not +35?
yes it is +35 sorry
The answer I get is x^3+5
To divide (x^6 - 12x^3 - 35) by (x^3 - 7), we can use polynomial long division. Here is the step-by-step process:
Step 1: Write the dividend (x^6 - 12x^3 - 35) and the divisor (x^3 - 7) in descending order of exponents:
____
x^3 - 7 | x^6 - 12x^3 - 35
Step 2: Divide the first term of the dividend (x^6) by the first term of the divisor (x^3) to get x^(6-3) = x^3. Write this as the first term of the quotient.
x^3
____
x^3 - 7 | x^6 - 12x^3 - 35
Step 3: Multiply the entire divisor (x^3 - 7) by the first term of the quotient (x^3).
x^3
____
x^3 - 7 | x^6 - 12x^3 - 35
- (x^6 - 7x^3)
Step 4: Subtract the result obtained in step 3 from the dividend (x^6 - 12x^3 - 35).
x^3
____
x^3 - 7 | x^6 - 12x^3 - 35
- (x^6 - 7x^3)
_____________
-5x^3
Step 5: Bring down the next term from the dividend (-5x^3).
x^3 - 5x^3
____
x^3 - 7 | x^6 - 12x^3 - 35
- (x^6 - 7x^3)
_____________
-5x^3 - 35
Step 6: Divide the first term of the new dividend (-5x^3) by the first term of the divisor (x^3) to get -5. Write this as the second term of the quotient.
x^3 - 5x^3
____
x^3 - 7 | x^6 - 12x^3 - 35
- (x^6 - 7x^3)
_____________
-5x^3 - 35
- (-5x^3 + 35)
Step 7: Multiply the entire divisor (x^3 - 7) by the second term of the quotient (-5).
x^3 - 5x^3
____
x^3 - 7 | x^6 - 12x^3 - 35
- (x^6 - 7x^3)
_____________
-5x^3 - 35
- (-5x^3 + 35)
____________
-70
Step 8: Subtract the result obtained in step 7 from the new dividend (-5x^3 - 35).
x^3 - 5x^3
____
x^3 - 7 | x^6 - 12x^3 - 35
- (x^6 - 7x^3)
_____________
-5x^3 - 35
- (-5x^3 + 35)
____________
-70
Step 9: Since there are no more terms left in the dividend, the division is complete.
Therefore, the quotient of (x^6 - 12x^3 - 35) divided by (x^3 - 7) is x^3 - 5x^3 - 10 with a remainder of -70.