%2F(x%2B5).jpeg "(x^4+4x^3+16x-35)/(x+5)")
Polynomial Long Division: (x^4 + 4x^3 + 16x - 35) / (x + 5)
This article will demonstrate how to perform polynomial long division to simplify the expression (x^4 + 4x^3 + 16x - 35) / (x + 5).
Step 1: Set Up the Division
Begin by setting up the long division problem:
________
x + 5 | x^4 + 4x^3 + 0x^2 + 16x - 35
Notice that we have included a placeholder term (0x^2) for the missing x^2 term in the dividend.
Step 2: Divide the Leading Terms
Divide the leading term of the dividend (x^4) by the leading term of the divisor (x). This gives us x^3:
x^3
x + 5 | x^4 + 4x^3 + 0x^2 + 16x - 35
Step 3: Multiply the Quotient by the Divisor
Multiply the quotient (x^3) by the divisor (x + 5):
x^3
x + 5 | x^4 + 4x^3 + 0x^2 + 16x - 35
-(x^4 + 5x^3)
Step 4: Subtract and Bring Down the Next Term
Subtract the result from the dividend and bring down the next term (0x^2):
x^3
x + 5 | x^4 + 4x^3 + 0x^2 + 16x - 35
-(x^4 + 5x^3)
-x^3 + 0x^2
Step 5: Repeat Steps 2-4
Repeat steps 2-4 until the degree of the remainder is less than the degree of the divisor.
Divide the leading term of the new dividend (-x^3) by the leading term of the divisor (x) to get -x^2:
x^3 - x^2
x + 5 | x^4 + 4x^3 + 0x^2 + 16x - 35
-(x^4 + 5x^3)
-x^3 + 0x^2
-(-x^3 - 5x^2)
Subtract and bring down the next term (16x):
x^3 - x^2
x + 5 | x^4 + 4x^3 + 0x^2 + 16x - 35
-(x^4 + 5x^3)
-x^3 + 0x^2
-(-x^3 - 5x^2)
5x^2 + 16x
Repeat the process:
x^3 - x^2 + 5x
x + 5 | x^4 + 4x^3 + 0x^2 + 16x - 35
-(x^4 + 5x^3)
-x^3 + 0x^2
-(-x^3 - 5x^2)
5x^2 + 16x
-(5x^2 + 25x)
-9x - 35
Finally:
x^3 - x^2 + 5x - 9
x + 5 | x^4 + 4x^3 + 0x^2 + 16x - 35
-(x^4 + 5x^3)
-x^3 + 0x^2
-(-x^3 - 5x^2)
5x^2 + 16x
-(5x^2 + 25x)
-9x - 35
-(-9x - 45)
10
Solution
The quotient is x^3 - x^2 + 5x - 9 and the remainder is 10. Therefore, the simplified expression can be written as:
(x^4 + 4x^3 + 16x - 35) / (x + 5) = x^3 - x^2 + 5x - 9 + 10/(x+5)