(6x3 – 12x + 10) ÷ (3x – 3)
To divide the expression (6x^3 - 12x + 10) by (3x - 3), we can use polynomial long division. Long division is a method used to divide polynomials term by term.
Step 1: Arrange the expression in descending order.
The expression (6x^3 - 12x + 10) by (3x - 3) can be written as:
6x^3 - 12x + 10
----------------
3x - 3
Step 2: Divide the first term of the dividend by the first term of the divisor.
In this case, the first term of the dividend is 6x^3 and the first term of the divisor is 3x. So, 6x^3 divided by 3x is 2x^2. Write this as the first term of the quotient above the line.
2x^2
---------------
3x - 3 | 6x^3 - 12x + 10
Step 3: Multiply the divisor by the quotient term and write it below the dividend.
Multiply (3x - 3) by 2x^2, which gives you 6x^3 - 6x^2.
2x^2
---------------
3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
Step 4: Subtract the product from the dividend.
Subtract (6x^3 - 6x^2) from (6x^3 - 12x + 10), which gives you -6x^2 - 12x + 10.
2x^2
---------------
3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
----------
- 6x^2 - 12x + 10
Step 5: Bring down the next term.
Bring down the next term, which is -6x^2.
2x^2 - 2
---------------
3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
----------
- 6x^2 - 12x + 10
- ( - 6x^2 + 6x)
Step 6: Divide the new term by the divisor.
In this case, -6x^2 divided by 3x is -2x. Write this as the next term of the quotient.
2x^2 - 2x
---------------
3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
----------
- 6x^2 - 12x + 10
- ( - 6x^2 + 6x)
---------------
- 18x + 10
Step 7: Multiply the divisor by the new quotient term and write it below.
Multiply (3x - 3) by -2x, which gives you -6x^2 + 6x.
2x^2 - 2x
---------------
3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
----------
- 6x^2 - 12x + 10
- ( - 6x^2 + 6x)
---------------
- 18x + 10
+ ( - 18x + 18)
Step 8: Subtract the product from the previous result.
Subtract (-6x^2 + 6x) from (-18x + 10), which gives you -24x + 10.
2x^2 - 2x - 6
---------------
3x - 3 | 6x^3 - 12x + 10
- (6x^3 - 6x^2)
-------------
- 6x^2 - 12x + 10
- ( - 6x^2 + 6x)
---------------
- 18x + 10
+ ( - 18x + 18)
-------------------
- 24x + 28
Step 9: The division is complete.
The final result of dividing (6x^3 - 12x + 10) by (3x - 3) is the quotient 2x^2 - 2x - 6 with a remainder of -24x + 28.
Therefore, the division is written as:
(6x^3 - 12x + 10) ÷ (3x - 3) = 2x^2 - 2x - 6 + (-24x + 28) / (3x - 3)