Which expression is equal to
(6x^3+10x^2-4x+10)/(3x-1)?
1.) 2x^2+4x+10+(10/3x-1)
2.) 2x^2+4x+(10/3x-1)
3.) (2x^2)+(8/3x)-(20/9)+(110/27x-9)
4.) 2x^2-4x-10+(10/3x-1)
Ignoring the rather casual use of parentheses, a little long division reveals the answer is (2)
To find the expression that is equal to (6x^3+10x^2-4x+10)/(3x-1), we need to perform polynomial division.
Let's go through the steps:
Step 1: Write the division problem in long division format:
_____________________
3x - 1 | 6x^3 + 10x^2 - 4x + 10
Step 2: Divide the first term of the dividend (6x^3) by the first term of the divisor (3x). The result is 2x^2.
Step 3: Multiply the entire divisor (3x - 1) by the quotient from Step 2 (2x^2) and write the result below the dividend, aligned under the like terms:
2x^2(3x - 1) = 6x^3 - 2x^2
Step 4: Subtract the result from Step 3 from the original dividend:
6x^3 + 10x^2 - 4x + 10
- (6x^3 - 2x^2)
_____________________
12x^2 - 4x + 10
Step 5: Bring down the next term (10) and repeat the process.
Step 6: Divide the first term (12x^2) by the first term of the divisor (3x). The result is 4x.
Step 7: Multiply the entire divisor (3x - 1) by the quotient from Step 6 (4x) and write the result below the previous result:
4x(3x - 1) = 12x^2 - 4x
Step 8: Subtract the result from Step 7 from the previous result:
12x^2 - 4x + 10
- (12x^2 - 4x)
___________________
10
Step 9: There are no more terms to bring down. The remainder is 10.
Finally, the expression (6x^3+10x^2-4x+10)/(3x-1) can be expressed as:
2x^2 + 4x + 10 + 10/(3x - 1)
Therefore, the correct answer is option 1.) 2x^2 + 4x + 10 + 10/(3x - 1).