Directions: Find the quotient and the remainder when the first polynomial is divided by the second.

#9. 3x^4 – 2x^3 + 5x^2 + x + 1;x^2 + 2x

Do exactly as you would do in a long division of numbers:

3x^4 – 2x^3 + 5x^2 + x + 1;x^2 + 2x

I will do it below, but the coefficients will not be lined up because I do not know how to insert spaces that don't get merged at this forum.
Divide x^2+2x into

3x^4 – 2x^3 + 5x^2 + x + 1

First divide 3x^4 by x^2 to get 3x^2.

Then multiply (x^2+2x) by 3x^2 to get
3x^4+6x^3

Subtract 3x^4+6x^3 from 3x^4 – 2x^3 + 5x^2 + x + 1 to get
– 8x^3 + 5x^2 + x + 1

Repeat the same as above:
Divide -8x^3 by x^2 to get -8x

Multiply x^2+2x by -8x to get -8x^3 -16x^2.

Subtract -8x^3 -16x^2 from – 8x^3 + 5x^2 + x + 1 to get 21x^2 +x + 1.

The same process will repeat itself to get the final answer by collecting all the above quotient terms:
3x^2-8x+21 with a remainder of -41x+1

To find the quotient and the remainder when the first polynomial is divided by the second, we can use long division.

1. Write the first polynomial (dividend) and the second polynomial (divisor) in descending order of powers of x:
Dividend: 3x^4 – 2x^3 + 5x^2 + x + 1
Divisor: x^2 + 2x

2. Divide the first term of the dividend (3x^4) by the first term of the divisor (x^2).
The result is 3x^2.

3. Multiply the divisor (x^2 + 2x) by the quotient (3x^2).
The product is 3x^4 + 6x^3.

4. Subtract the product (3x^4 + 6x^3) from the dividend (3x^4 – 2x^3 + 5x^2 + x + 1).
The result is -8x^3 + 5x^2 + x + 1.

5. Bring down the next term (-8x^3) from the dividend.

6. Divide the first term of the new dividend (-8x^3) by the first term of the divisor (x^2).
The result is -8x.

7. Multiply the divisor (x^2 + 2x) by the new quotient (-8x).
The product is -8x^3 - 16x^2.

8. Subtract the product (-8x^3 - 16x^2) from the new dividend (-8x^3 + 5x^2 + x + 1).
The result is 21x^2 + x + 1.

9. Now, we only have terms of degree 2 or less in the dividend.
Divide the first term of the new dividend (21x^2) by the first term of the divisor (x^2).
The result is 21.

10. Multiply the divisor (x^2 + 2x) by the new quotient (21).
The product is 21x^2 + 42x.

11. Subtract the product (21x^2 + 42x) from the new dividend (21x^2 + x + 1).
The result is -41x + 1.

12. Finally, since we have no more terms to bring down, the remainder is -41x + 1.

The quotient is 3x^2 - 8x + 21 and the remainder is -41x + 1.

To find the quotient and the remainder when the first polynomial is divided by the second, we can use polynomial long division.

First, arrange the polynomial in descending order of exponents:
3x^4 – 2x^3 + 5x^2 + x + 1

Next, divide the first term of the dividend (3x^4) by the first term of the divisor (x^2) to obtain the first term of the quotient. In this case, 3x^4 divided by x^2 equals 3x^2. Write this term above the line:

__________
x^2 + 2x | 3x^4 – 2x^3 + 5x^2 + x + 1
3x^2

Next, multiply the entire divisor (x^2 + 2x) by the first term of the quotient (3x^2) and subtract the result from the dividend. Write this subtraction result underneath the dividend:

__________
x^2 + 2x | 3x^4 – 2x^3 + 5x^2 + x + 1
-(3x^4 + 6x^3)

Next, bring down the next term of the dividend (5x^2) and repeat the process:

__________
x^2 + 2x | 3x^4 – 2x^3 + 5x^2 + x + 1
-(3x^4 + 6x^3)
-----------
-8x^3 + 5x^2

Divide the first term of the resulting polynomial (-8x^3) by the first term of the divisor (x^2). In this case, -8x^3 divided by x^2 equals -8x. Write this term above the line:

__________
x^2 + 2x | 3x^4 – 2x^3 + 5x^2 + x + 1
-(3x^4 + 6x^3)
-----------
-8x^3 + 5x^2
-8x^3 - 16x^2

Repeat the process by multiplying the entire divisor (x^2 + 2x) by the current term of the quotient (-8x) and subtracting the result from the previous subtraction:

__________
x^2 + 2x | 3x^4 – 2x^3 + 5x^2 + x + 1
-(3x^4 + 6x^3)
-----------
-8x^3 + 5x^2
-8x^3 - 16x^2
------------------
21x^2 + x

Next, bring down the next term of the dividend (x) and repeat the process:

__________
x^2 + 2x | 3x^4 – 2x^3 + 5x^2 + x + 1
-(3x^4 + 6x^3)
-----------
-8x^3 + 5x^2
-8x^3 - 16x^2
------------------
21x^2 + x
21x^2 + 42x
---------------------
-41x + 1

Lastly, bring down the last term of the dividend (1) and repeat the process:

__________
x^2 + 2x | 3x^4 – 2x^3 + 5x^2 + x + 1
-(3x^4 + 6x^3)
-----------
-8x^3 + 5x^2
-8x^3 - 16x^2
------------------
21x^2 + x
21x^2 + 42x
---------------------
-41x + 1
-41x - 82
-------------------------
83

Therefore, the quotient when 3x^4 – 2x^3 + 5x^2 + x + 1 is divided by x^2 + 2x is 3x^2 - 8x + 21, and the remainder is 83.