Consider the polynomial division (x^3+5x^2+4x+18)/(x+5). Give the quotient with the remainder expressed as a fraction.
-27/(x+5) is not the answer
To divide the polynomial (x^3+5x^2+4x+18) by (x+5), we can use long division.
x^2 - 4x + 16
___________________________________
x + 5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
_____________
- 4x + 18
- (- 4x + 20)
_____________
- 2
The quotient is x^2 - 4x + 16, and the remainder is -2.
Therefore, the polynomial division (x^3+5x^2+4x+18)/(x+5) can be expressed as:
x^2 - 4x + 16 - 2/(x+5)
To find the quotient with the remainder expressed as a fraction, we can use polynomial long division.
Step 1: Set up the division as follows:
________________________
(x + 5) | x^3 + 5x^2 + 4x + 18
Step 2: Divide the first term of the dividend (x^3) by the first term of the divisor (x), which gives us x^2.
________________________
(x + 5) | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
-----------
0 + 4x + 18
Step 3: Bring down the next term (4x) and divide it by the first term of the divisor (x), which gives us 4.
_________________________
(x + 5) | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
-----------
4x + 18
- (4x + 20)
---------------
-2
Step 4: Since we have no terms with a degree higher than 1, we have our remainder as -2.
Therefore, the quotient is x^2 + 4, and the remainder expressed as a fraction is -2/(x + 5).
To divide the polynomial (x^3+5x^2+4x+18) by (x+5), you can follow these steps for polynomial long division:
Step 1: Divide the highest exponent term of the numerator by the highest exponent term of the denominator.
Divide (x^3) by (x) to get x^2.
Step 2: Multiply the whole denominator with the quotient obtained in the previous step.
Multiply (x+5) by (x^2) to get (x^3+5x^2).
Step 3: Subtract the product obtained in the previous step from the numerator.
Subtract (x^3+5x^2+4x+18) - (x^3+5x^2) to get (4x+18).
Step 4: Bring down the next term from the numerator.
Bring down the term (4x) to get (4x+18).
Step 5: Repeat steps 1 to 4 until there are no more terms in the numerator.
Step 1: Divide (4x) by (x) to get 4.
Step 2: Multiply (x+5) by 4 to get (4x+20).
Step 3: Subtract (4x+18) - (4x+20) to get -2.
Since there are no more terms remaining in the numerator, the quotient becomes x^2 + 4 and the remainder is -2.
Therefore, the polynomial division (x^3+5x^2+4x+18)/(x+5) is equal to the quotient x^2 + 4, with the remainder expressed as a fraction -2/(x+5).