Divide using long division. Check your answers.

Question

(x^3 + 7x^2 + 15x + 9) / (x +1)

Answer 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.