divide: (12x^3-42x^2-30x-3)/(6x+3)
To divide the polynomial (12x^3 - 42x^2 - 30x - 3) by (6x + 3), you can use the long division method. Here's how you can do it step by step:
Step 1: Ensure the polynomial is written in descending order of degree.
The given polynomial (12x^3 - 42x^2 - 30x - 3) is already in descending order, so we can move on to the next step.
Step 2: Divide the first term of the dividend by the first term of the divisor.
Divide 12x^3 by 6x, which gives us 2x^2.
2x^2
Step 3: Multiply the divisor by the quotient obtained in the previous step.
Multiply (6x + 3) by 2x^2, which gives us 12x^3 + 6x^2.
2x^2
___________
6x + 3 | 12x^3 - 42x^2 - 30x - 3
-(12x^3 + 6x^2)
___________
-36x^2 - 30x - 3
Step 4: Subtract the result obtained in the previous step (12x^3 + 6x^2) from the dividend (12x^3 - 42x^2 - 30x - 3).
(12x^3 - 42x^2 - 30x - 3) - (12x^3 + 6x^2) = -36x^2 - 30x - 3
2x^2
___________
6x + 3 | 12x^3 - 42x^2 - 30x - 3
-(12x^3 + 6x^2)
___________
-36x^2 - 30x - 3
Step 5: Repeat steps 2-4 until there are no more terms to divide.
Now, repeat the process using the new dividend (-36x^2 - 30x - 3).
Step 6: Divide the first term of the new dividend (-36x^2) by the first term of the divisor (6x).
Divide -36x^2 by 6x, which gives us -6x.
2x^2 - 6x
Step 7: Multiply the divisor by the new quotient (-6x).
Multiply (6x + 3) by -6x, which gives us -36x^2 - 18x.
2x^2 - 6x
___________
6x + 3 | 12x^3 - 42x^2 - 30x - 3
-(12x^3 + 6x^2)
___________
-36x^2 - 30x - 3
+36x^2 + 18x
___________
-12x - 3
Step 8: Subtract the result obtained in the previous step (-36x^2 - 18x) from the new dividend (-36x^2 - 30x - 3).
(-36x^2 - 30x - 3) - (-36x^2 - 18x) = -12x - 3
2x^2 - 6x
___________
6x + 3 | 12x^3 - 42x^2 - 30x - 3
-(12x^3 + 6x^2)
___________
-36x^2 - 30x - 3
+36x^2 + 18x
___________
-12x - 3
Step 9: Verify if there are any remaining terms to divide.
In this case, we have a remaining term (-12x - 3) in the dividend. Since this term has a lesser degree than the divisor (6x + 3), we cannot divide any further.
Step 10: Write the final quotient and remainder.
The final quotient is (2x^2 - 6x) and the remainder is (-12x - 3). Therefore,
(12x^3 - 42x^2 - 30x - 3) divided by (6x + 3) is equal to (2x^2 - 6x) with a remainder of (-12x - 3).
Final Result:
Quotient = 2x^2 - 6x
Remainder = -12x - 3