Consider the polynomial division (x^3+5x^2+4x+18)/(x+5). Give the quotient with the remainder expressed as a fraction.
You were wrong last time
I apologize for any confusion caused. To perform polynomial division, let's divide (x^3+5x^2+4x+18) by (x+5).
- x^2 + 6x - 10
_______________________
x + 5 │ x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
________________
-x^2 + 4x
+ ( -x^2 - 5x)
________________
9x + 18
- (9x + 45)
_______________
-27
The quotient is -x^2 + 6x - 10 and the remainder is -27. Therefore, the polynomial division (x^3+5x^2+4x+18)/(x+5) gives the quotient as -x^2 + 6x - 10, with the remainder expressed as -27/ (x+5).
Apologies for any confusion caused in my previous response. Let's proceed with the correct solution.
To divide the polynomial (x^3+5x^2+4x+18) by (x+5), we can use long division. Here are the steps:
Step 1: Divide the first term of the dividend (x^3) by the first term of the divisor (x). The result is x^2.
x^2
___________
x + 5 | x^3 + 5x^2 + 4x + 18
Step 2: Multiply the divisor (x+5) by the result obtained in Step 1 (x^2). The result is x^2(x+5) = x^3 + 5x^2.
x^3 + 5x^2
___________
x + 5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
_________________
- 4x^2
Step 3: Subtract the term obtained in Step 2 (x^3 + 5x^2) from the dividend. The result is -4x^2.
x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
_________________
- 4x^2
Step 4: Bring down the next term from the dividend (-4x). The new dividend is -4x^2 + 4x.
x^2 + (-4x)
___________
x + 5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
_________________
- 4x^2
- (-4x^2 + 4x)
________________
8x
Step 5: Divide the new term of the dividend (8x) by the first term of the divisor (x). The result is 8.
x^2 + (-4x) + 8
___________
x + 5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
_________________
- 4x^2
- (-4x^2 + 4x)
________________
8x
- (8x + 40)
_________________
- 22
Step 6: Bring down the last term from the dividend (18). The new dividend is -22 + 18 = -4.
x^2 + (-4x) + 8 - (22/(x+5))
___________
x + 5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
_________________
- 4x^2
- (-4x^2 + 4x)
________________
8x
- (8x + 40)
_________________
- 22
-22 + 18
__________
-4
The quotient, with the remainder expressed as a fraction, is:
x^2 - 4x + 8 - (22/(x+5))
Please let me know if I can help you with anything else.
My apologies for any previous incorrect response. Let's find the quotient and remainder for the polynomial division.
To divide the polynomial (x^3+5x^2+4x+18) by (x+5), we can perform long division. Here's how:
________________________
x+5 | x^3 + 5x^2 + 4x + 18
First, divide x^3 by x to get x^2. Write x^2 above the line.
x^2
________________________
x+5 | x^3 + 5x^2 + 4x + 18
Multiply (x^2) by (x+5) to get (x^3+5x^2). Write this term below the dividend and subtract from the previous line:
x^2
________________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
This leaves us with (4x + 18). Bring down the next term, 4x.
x^2
________________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
-----------
4x + 18
Now divide (4x) by (x+5) to get 4. Write 4 above the line.
x^2 + 4
________________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
-----------
4x + 18
Multiply (4) by (x+5) to get (4x+20). Write this term below the previous subtraction and subtract:
x^2 + 4
________________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
-----------
4x + 18
- (4x + 20)
-----------
-2
At this point, we have a remainder of -2. Since the divisor is (x+5), the remainder can be expressed as a fraction by dividing the remainder (-2) by the divisor (x+5):
Remainder = -2
Divisor = (x+5)
Therefore, the remainder expressed as a fraction is: -2/(x+5).
The quotient is x^2 + 4 and the remainder expressed as a fraction is -2/(x+5).