Question 149767
Let's simplify this expression using synthetic division



Start with the given expression {{{(x^7 - x^6 + x^5 - x^4 + 4)/(x-1)}}}


First lets find our test zero:


{{{x-1=0}}} Set the denominator {{{x-1}}} equal to zero


{{{x=1}}} Solve for x.


so our test zero is 1



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 {{{-x^4}}} to {{{4}}} there is a zero coefficient for {{{x^3}}}, {{{x^2}}}, and {{{x}}}. This is simply because {{{x^7 - x^6 + x^5 - x^4 + 4}}} really looks like {{{1x^7+-1x^6+1x^5+-1x^4+0x^3+0x^2+0x^1+4x^0}}}<TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><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><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>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><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><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</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><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</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>0</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD></TD><TD></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><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>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD></TD><TD></TD><TD></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><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD></TD><TD></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><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</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>0</TD><TD>0</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</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>0</TD><TD>0</TD><TD>0</TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD></TD><TD></TD></TR></TABLE>

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

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD></TD></TR></TABLE>

    Multiply 1 by 0 and place the product (which is 0)  right underneath the eighth  coefficient (which is 4)

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD></TD></TR></TABLE>

    Add 0 and 4 to get 4. Place the sum right underneath 0.

    <TABLE cellpadding=10><TR><TD>1</TD><TD>|</TD><TD>1</TD><TD>-1</TD><TD>1</TD><TD>-1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>1</TD><TD>0</TD><TD>1</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>0</TD><TD>4</TD></TR></TABLE>



Since the last column adds to 4, we have a remainder of 4. 



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Answer:


So when {{{x^7-x^6+x^5-x^4+4}}} is divided by {{{x-1}}}, the remainder is 4