divide.x^4+x^2-80/x+3
-3|1 0 1 0 -80
...... -3 9-30 90
.....1-3 10-30 10
the above is to show the algorithm of synthetic division. It is hard to show a nice matrix in this forum
my answer represents x^3 - 3x^2 + 10x - 30 with a remainder of 10
the following page shows how to do it using long algebraic division
forgot to post the webpage
http://www.sosmath.com/algebra/factor/fac01/fac01.html
To divide the expression (x^4 + x^2 - 80) by (x + 3), we can use the polynomial long division method. Here's how to do it step by step:
Step 1: Arrange the dividend (x^4 + x^2 - 80) and the divisor (x + 3) in the long division format:
_____________________
x + 3 | x^4 + x^2 - 80
Step 2: Determine the term to divide into the first term of the dividend (x^4) to obtain the first term of the quotient. In this case, it is (x^4 ÷ x) = x^3. Place this term above the horizontal line:
x^3
_______
x + 3 | x^4 + x^2 - 80
Step 3: Multiply the divisor (x + 3) by the term we just found (x^3). Write the result below the dividend, then subtract it from the dividend:
x^3
_______
x + 3 | x^4 + x^2 - 80
- (x^4 + 3x^3)
This subtraction leaves us with:
x^4 + x^2 - 80
- (x^4 + 3x^3)
________________
-3x^3 + x^2
Step 4: Bring down the next term from the dividend (x^2) and bring it directly beneath the previous result:
x^3
_______
x + 3 | x^4 + x^2 - 80
- (x^4 + 3x^3)
________________
-3x^3 + x^2
- ( -3x^3 - 9x^2)
This subtraction gives us:
x^3
_______
x + 3 | x^4 + x^2 - 80
- (x^4 + 3x^3)
________________
-3x^3 + x^2
- ( -3x^3 - 9x^2)
__________________________
10x^2 - 80
Step 5: Repeat the process by determining the term to divide into the next term of the result (-3x^3 + x^2). In this case, it is (-3x^3 ÷ x) = -3x^2. Place this term above the horizontal line:
x^3 - 3x^2
_____________
x + 3 | x^4 + x^2 - 80
Step 6: Multiply the divisor (x + 3) by the new term (-3x^2). Write the result below the previous result, then subtract it from the previous result:
x^3 - 3x^2
_____________
x + 3 | x^4 + x^2 - 80
- (x^4 + 3x^3)
________________
-3x^3 + x^2
- ( -3x^3 - 9x^2)
__________________________
10x^2 - 80
- (10x^2 - 30x)
This subtraction gives us:
x^3 - 3x^2
_____________
x + 3 | x^4 + x^2 - 80
- (x^4 + 3x^3)
________________
-3x^3 + x^2
- ( -3x^3 - 9x^2)
__________________________
10x^2 - 80
- (10x^2 - 30x)
_________________________________
30x - 80
Step 7: Bring down the next term from the dividend (-80) and bring it directly beneath the previous result:
x^3 - 3x^2 + 30x
__________________
x + 3 | x^4 + x^2 - 80
- (x^4 + 3x^3)
________________
-3x^3 + x^2
- ( -3x^3 - 9x^2)
__________________________
10x^2 - 80
- (10x^2 - 30x)
_________________________________
30x - 80
- (30x + 90)
This subtraction gives us:
x^3 - 3x^2 + 30x
__________________
x + 3 | x^4 + x^2 - 80
- (x^4 + 3x^3)
________________
-3x^3 + x^2
- ( -3x^3 - 9x^2)
__________________________
10x^2 - 80
- (10x^2 - 30x)
_________________________________
30x - 80
- (30x + 90)
__________________________________________
- 170
Step 8: The remainder we obtained from the last subtraction is -170. Since we have completed all the terms from the dividend, this is the final result.
Therefore, (x^4 + x^2 - 80) ÷ (x + 3) = x^3 - 3x^2 + 30x - (170 / (x + 3))