Divide using long division. Check your answers.
(x^3 + 7x^2 + 15x + 9) / (x +1)
To divide using long division, we start by dividing the first term of the dividend (x^3) by the divisor (x + 1).
_______
x + 1 | x^3 + 7x^2 + 15x + 9
We can see that x^3 divided by x gives us x^2. So, let's write x^2 above the line:
x^2
_______
x + 1 | x^3 + 7x^2 + 15x + 9
Now, we need to multiply the divisor (x + 1) by the quotient (x^2) to get a new dividend.
x^2
_________________
x + 1 | x^3 + 7x^2 + 15x + 9
-(x^3 + x^2)
By subtracting (x^3 + x^2) from (x^3 + 7x^2), we get 6x^2:
x^2 + 6x^2
_________________
x + 1 | x^3 + 7x^2 + 15x + 9
-(x^3 + x^2)
_____________
6x^2 + 15x + 9
Now, we bring down the next term of the dividend, which is 15x:
x^2 + 6x^2
_________________
x + 1 | x^3 + 7x^2 + 15x + 9
-(x^3 + x^2)
_____________
6x^2 + 15x + 9
-(6x^2 + 15x)
By subtracting (6x^2 + 15x) from (6x^2 + 15x + 9), we get 9:
x^2 + 6x^2 + 9
_________________
x + 1 | x^3 + 7x^2 + 15x + 9
-(x^3 + x^2)
_____________
6x^2 + 15x + 9
-(6x^2 + 15x)
__________
9
Since 9 is the constant term and there are no more terms in the dividend, our final quotient is x^2 + 6x^2 + 9.
To check our answer, we can multiply the quotient by the divisor and add the remainder:
(x + 1)(x^2 + 6x^2 + 9) + 9
This simplifies to:
x^3 + 7x^2 + 15x + 9
Thus, our answer is correct. The quotient is x^2 + 6x^2 + 9 and the remainder is 9.