Consider the polynomial division (x^3+5x^2+4x+18)/(x+5). Give the quotient with the remainder expressed as a fraction.
To find the quotient with the remainder, we can use polynomial long division.
x^2 - 4x + 14
___________________
x + 5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
_______________
-x^2 + 4x
+ (-x^2 - 5x)
_______________
9x + 18
- (9x + 45)
_______________
-27
Therefore, the quotient is x^2 - 4x + 14 and the remainder expressed as a fraction is -27/(x + 5).
To perform polynomial division, we divide the polynomial (x^3+5x^2+4x+18) by the binomial (x+5).
First, we divide the leading term of the polynomial by the leading term of the binomial:
(x^3) / (x) = x^2
Next, we multiply the binomial (x+5) by the quotient we found:
(x+5) * (x^2) = x^3 + 5x^2
Subtract this product from the original polynomial:
(x^3 + 5x^2 + 4x + 18) - (x^3 + 5x^2) = 4x + 18
Now, we divide the resulting polynomial (4x + 18) by the binomial (x+5):
(4x) / (x) = 4
Multiply the binomial (x+5) by the quotient we found:
(x+5) * 4 = 4x + 20
Subtract this product from the remaining polynomial:
(4x + 18) - (4x + 20) = -2
Therefore, the quotient is x^2 + 4, and the remainder is -2.
Expressing the remainder as a fraction, we have:
-2 / (x+5)
To perform polynomial division, we need to divide the polynomial (x^3+5x^2+4x+18) by the polynomial (x+5).
Step 1: Write the dividend (x^3+5x^2+4x+18) as a long division problem, with the dividend inside the long division symbol and the divisor (x+5) outside the symbol:
____________________
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) to get x^2. Place this result above the long division line:
x^2
____________________
x+5 | x^3 + 5x^2 + 4x + 18
Step 3: Multiply the divisor (x+5) by the quotient term (x^2), and write the result below the dividend term you just divided (x^3). Subtract this new polynomial from the corresponding terms of the dividend:
x^2
____________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
Simplified form: -x^2 + 4x + 18
Step 4: Bring down the next term of the dividend (-x^2). Now, divide this term by the first term of the divisor (x) to obtain -x. Place this new quotient term (-x) above the long division line:
x^2 - x
____________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
__________________
-x^2 + 4x
Step 5: Multiply the divisor (x+5) by the new quotient term (-x) and subtract the result from the simplified polynomial (-x^2 + 4x):
x^2 - x
____________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
__________________
-x^2 + 4x
- (-x^2 - 5x)
Simplified form: 9x + 18
Step 6: Bring down the next term of the dividend (9x). Divide this term by the first term of the divisor (x) to get 9. Place this result above the long division line:
x^2 - x + 9
____________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
__________________
-x^2 + 4x
- (-x^2 - 5x)
__________________
9x + 18
Step 7: Multiply the divisor (x+5) by the quotient term (9) and subtract it from the previous simplified polynomial:
x^2 - x + 9
____________________
x+5 | x^3 + 5x^2 + 4x + 18
- (x^3 + 5x^2)
__________________
-x^2 + 4x
- (-x^2 - 5x)
__________________
9x + 18
- (9x + 45)
Simplified form: -27
Step 8: Since there are no more terms to bring down, we can conclude that the remainder is -27.
Therefore, the quotient with the remainder expressed as a fraction is (x^2 - x + 9) with a remainder of -27.