Question 90278


Start with the given polynomial {{{(x^4 - 2x^3 + x^2 - 3x + 2)/(x-2)}}}


First lets find our test zero:


{{{x-2=0}}} Set the denominator {{{x-2}}} equal to zero

{{{x=2}}} Solve for x.


so our test zero is 2



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.<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>1</TD><TD>-2</TD><TD>1</TD><TD>-3</TD><TD>2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

<TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>1</TD><TD>-2</TD><TD>1</TD><TD>-3</TD><TD>2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 2 by 1 and place the product (which is 2)  right underneath the second  coefficient (which is -2)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>1</TD><TD>-2</TD><TD>1</TD><TD>-3</TD><TD>2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>2</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 2 and -2 to get 0. Place the sum right underneath 2.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>1</TD><TD>-2</TD><TD>1</TD><TD>-3</TD><TD>2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>2</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 2 by 0 and place the product (which is 0)  right underneath the third  coefficient (which is 1)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>1</TD><TD>-2</TD><TD>1</TD><TD>-3</TD><TD>2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>2</TD><TD>0</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add 0 and 1 to get 1. Place the sum right underneath 0.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>1</TD><TD>-2</TD><TD>1</TD><TD>-3</TD><TD>2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>2</TD><TD>0</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply 2 by 1 and place the product (which is 2)  right underneath the fourth  coefficient (which is -3)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>1</TD><TD>-2</TD><TD>1</TD><TD>-3</TD><TD>2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>2</TD><TD>0</TD><TD>2</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD></TD><TD></TD></TR></TABLE>

    Add 2 and -3 to get -1. Place the sum right underneath 2.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>1</TD><TD>-2</TD><TD>1</TD><TD>-3</TD><TD>2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>2</TD><TD>0</TD><TD>2</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>-1</TD><TD></TD></TR></TABLE>

    Multiply 2 by -1 and place the product (which is -2)  right underneath the fifth  coefficient (which is 2)

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>1</TD><TD>-2</TD><TD>1</TD><TD>-3</TD><TD>2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>2</TD><TD>0</TD><TD>2</TD><TD>-2</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>-1</TD><TD></TD></TR></TABLE>

    Add -2 and 2 to get 0. Place the sum right underneath -2.

    <TABLE cellpadding=10><TR><TD>2</TD><TD>|</TD><TD>1</TD><TD>-2</TD><TD>1</TD><TD>-3</TD><TD>2</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>2</TD><TD>0</TD><TD>2</TD><TD>-2</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>-1</TD><TD>0</TD></TR></TABLE>

Since the last column adds to zero, we have a remainder of zero. This means {{{x-2}}} is a factor of  {{{x^4 - 2x^3 + x^2 - 3x + 2}}}


Now lets look at the bottom row of coefficients:


The first 4 coefficients (1,0,1,-1) form the quotient


{{{x^3 + x - 1}}}



So {{{(x^4 - 2x^3 + x^2 - 3x + 2)/(x-2)=x^3 + x - 1}}}



So in other words, you are correct.



You can use this <a href=http://calc101.com/webMathematica/long-divide.jsp>online polynomial division calculator</a> to check your work