SOLUTION: (4x^6+20x^5-24x^4-3x^2-13x+30/(x+6)

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: (4x^6+20x^5-24x^4-3x^2-13x+30/(x+6)      Log On


   

Question 149941: (4x^6+20x^5-24x^4-3x^2-13x+30/(x+6)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Let's simplify this expression using synthetic division


Start with the given expression %284x%5E6+%2B+20x%5E5+-+24x%5E4+-+3x%5E2+-+13x+%2B+30%29%2F%28x%2B6%29

First lets find our test zero:

x%2B6=0 Set the denominator x%2B6 equal to zero

x=-6 Solve for x.

so our test zero is -6


Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.(note: remember if a polynomial goes from -24x%5E4 to -3x%5E2 there is a zero coefficient for x%5E3. This is simply because 4x%5E6+%2B+20x%5E5+-+24x%5E4+-+3x%5E2+-+13x+%2B+30 really looks like 4x%5E6%2B20x%5E5%2B-24x%5E4%2B0x%5E3%2B-3x%5E2%2B-13x%5E1%2B30x%5E0
-6|420-240-3-1330
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 4)
-6|420-240-3-1330
|
4

Multiply -6 by 4 and place the product (which is -24) right underneath the second coefficient (which is 20)
-6|420-240-3-1330
|-24
4

Add -24 and 20 to get -4. Place the sum right underneath -24.
-6|420-240-3-1330
|-24
4-4

Multiply -6 by -4 and place the product (which is 24) right underneath the third coefficient (which is -24)
-6|420-240-3-1330
|-2424
4-4

Add 24 and -24 to get 0. Place the sum right underneath 24.
-6|420-240-3-1330
|-2424
4-40

Multiply -6 by 0 and place the product (which is 0) right underneath the fourth coefficient (which is 0)
-6|420-240-3-1330
|-24240
4-40

Add 0 and 0 to get 0. Place the sum right underneath 0.
-6|420-240-3-1330
|-24240
4-400

Multiply -6 by 0 and place the product (which is 0) right underneath the fifth coefficient (which is -3)
-6|420-240-3-1330
|-242400
4-400

Add 0 and -3 to get -3. Place the sum right underneath 0.
-6|420-240-3-1330
|-242400
4-400-3

Multiply -6 by -3 and place the product (which is 18) right underneath the sixth coefficient (which is -13)
-6|420-240-3-1330
|-24240018
4-400-3

Add 18 and -13 to get 5. Place the sum right underneath 18.
-6|420-240-3-1330
|-24240018
4-400-35

Multiply -6 by 5 and place the product (which is -30) right underneath the seventh coefficient (which is 30)
-6|420-240-3-1330
|-24240018-30
4-400-35

Add -30 and 30 to get 0. Place the sum right underneath -30.
-6|420-240-3-1330
|-24240018-30
4-400-350



Since the last column adds to zero, we have a remainder of zero. This means x%2B6 is a factor of 4x%5E6+%2B+20x%5E5+-+24x%5E4+-+3x%5E2+-+13x+%2B+30

Now lets look at the bottom row of coefficients:

The first 6 coefficients (4,-4,0,0,-3,5) form the quotient

4x%5E5+-+4x%5E4+-+3x+%2B+5


So