SOLUTION: Simplify: (x^4 + 10x^3 - 23x - 348x - 540) / (x + 5)

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Start with the given polynomial

First lets find our test zero:

Set the denominator equal to zero

Solve for x.

so our test zero is -5


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 to there is a zero coefficient for . This is simply because really looks like

-5	|	1	10	0	-348	-540
	|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)

-5	|	1	10	0	-348	-540
	|
		1

Multiply -5 by 1 and place the product (which is -5) right underneath the second coefficient (which is 10)

-5	|	1	10	0	-348	-540
	|		-5
		1

Add -5 and 10 to get 5. Place the sum right underneath -5.

-5	|	1	10	0	-348	-540
	|		-5
		1	5

Multiply -5 by 5 and place the product (which is -25) right underneath the third coefficient (which is 0)

-5	|	1	10	0	-348	-540
	|		-5	-25
		1	5

Add -25 and 0 to get -25. Place the sum right underneath -25.

-5	|	1	10	0	-348	-540
	|		-5	-25
		1	5	-25

Multiply -5 by -25 and place the product (which is 125) right underneath the fourth coefficient (which is -348)

-5	|	1	10	0	-348	-540
	|		-5	-25	125
		1	5	-25

Add 125 and -348 to get -223. Place the sum right underneath 125.

-5	|	1	10	0	-348	-540
	|		-5	-25	125
		1	5	-25	-223

Multiply -5 by -223 and place the product (which is 1115) right underneath the fifth coefficient (which is -540)

-5	|	1	10	0	-348	-540
	|		-5	-25	125	1115
		1	5	-25	-223

Add 1115 and -540 to get 575. Place the sum right underneath 1115.

-5	|	1	10	0	-348	-540
	|		-5	-25	125	1115
		1	5	-25	-223	575

Since the last column adds to 575, we have a remainder of 575. This means is not a factor of
Now lets look at the bottom row of coefficients:

The first 4 coefficients (1,5,-25,-223) form the quotient



and the last coefficient 575, is the remainder, which is placed over like this





Putting this altogether, we get:



So

which looks like this in remainder form:
remainder 575


You can use this online polynomial division calculator to check your work