Final Answer:
(3x²-10x²+7x+10) ÷ (3x+2) simplifies to (x-5)(3x+2).
Step-by-step explanation:
Here's the step-by-step simplification:
Identify the type of expression:
We have a polynomial expression divided by another polynomial expression.
Check for factoring opportunities:
Neither the numerator nor the denominator can be factored further.
Use polynomial division:
We can use polynomial long division or synthetic division to perform the division.
Polynomial Long Division:
x - 5
3x + 2 | 3x^2 - 10x^2 + 7x + 10
- (3x^2 + 2x)
------------
-12x^2 + 7x + 10
- ( -12x^2 - 8x )
------------
15x + 10
- ( 15x + 10 )
------------
0
Synthetic Division:
x - 5
3 2
3 -10 7 10
\ 9 -2 4
--------------------------
3 -1 5 14
Interpret the result:
The quotient obtained after the division is x - 5 and the remainder is 14. Therefore, the simplified expression is:
(3x²-10x²+7x+10) ÷ (3x+2) = x - 5 + 14 / (3x+2)
Since the remainder isn't zero, the final simplified form is:
(x - 5)(3x+2)