Consider the polynomial division (x^3+5x^2+4x+18)÷(x+5)
. Give the quotient with the remainder expressed as a fraction.
To divide the polynomial (x^3 + 5x^2 + 4x + 18) by (x + 5), we use long division:
x^2 - 4x + 16
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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 - 4x + 16 and the remainder is -27. So, the division (x^3 + 5x^2 + 4x + 18) ÷ (x + 5) can be expressed as:
(x^2 - 4x + 16) - 27 / (x + 5)
To perform the polynomial division (x^3+5x^2+4x+18) ÷ (x+5), we divide the polynomial inside the dividend term by the divisor term.
Dividing x+5 into x^3+5x^2+4x+18:
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x+5 | x^3 + 5x^2 + 4x + 18
First, we divide x into x^3, which gives us x^2. We then multiply x^2 by the divisor (x+5), which gives us x^3 + 5x^2.
x^2
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x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
______________
- 4x^2
Next, we subtract (x^3 + 5x^2) from (x^3 + 5x^2 + 4x + 18), which leaves us with -4x^2.
We bring down the next term, which is +4x, and continue the process:
x^2 - 4
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x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
______________
- 4x^2 + 4x
Next, we divide x into -4x^2, which gives us -4x. We multiply -4x by the divisor (x+5), giving us -4x^2 - 20x.
x^2 - 4
_______________________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
______________
- 4x^2 + 4x
- (- 4x^2 - 20x)
__________________
24x + 18
We subtract (-4x^2 - 20x) from (-4x^2 + 4x), which gives us 24x. We bring down the next term, +18, and continue the process:
x^2 - 4 + (24x / (x+5))
__________________________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
______________
- 4x^2 + 4x
- (- 4x^2 - 20x)
__________________
24x + 18
-(24x + 120)
__________
-102
Since we cannot divide (-102) by (x+5) anymore (as x^3 is no longer present in the dividend), we have reached the remainder. Therefore, the quotient with the remainder expressed as a fraction is:
Quotient: x^2 - 4 + (24x / (x+5))
Remainder: -102 / (x+5)
To find the quotient and remainder of the polynomial division, you need to use the long division method. Here's how you can do it:
Step 1: Write the dividend (x^3 + 5x^2 + 4x + 18) and divisor (x + 5) in the long division format.
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x + 5 | x^3 + 5x^2 + 4x + 18
Step 2: Divide the first term of the dividend (x^3) by the divisor (x). The result is x^2, which is written at the top.
x^2
_____________________
x + 5 | x^3 + 5x^2 + 4x + 18
Step 3: Multiply the divisor (x + 5) by the quotient (x^2), and write the result below the dividend.
x^2
_____________________
x + 5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
-4x^2 + 4x
Step 4: Bring down the next term of the dividend (-4x), and add it to the result obtained in the previous step.
x^2 - 4x
_____________________
x + 5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
_____________________
-4x^2 + 4x
-4x^2 - 20x
Step 5: Divide the result obtained in the previous step (-4x^2 - 20x) by the divisor (x + 5). The result is -4x, which is written at the top.
x^2 - 4x
_____________________
x + 5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
_____________________
-4x^2 + 4x
-4x^2 - 20x
- (-4x^2 - 20x)
24x + 18
Step 6: Bring down the last term of the dividend (24x) and add it to the result obtained in the previous step.
x^2 - 4x + 24x
_____________________
x + 5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
_____________________
-4x^2 + 4x
-4x^2 - 20x
- (-4x^2 - 20x)
_____________________
24x + 18
- (24x + 120)
-102
Step 7: The remainder obtained is -102.
Therefore, the quotient is x^2 - 4x + 24 and the remainder is -102, which can be expressed as a fraction as -102/1.