Question 970032
This mostly means the use of synthetic division to check if {{{-4}}} is a root of f(x).  Either remainder is zero or it is nonzero.  The dividend to use in the division must be  according to {{{f(x)=4x^6+0x^5-64x^4+0x^3+x^2+0*x-18}}}.


In case you are not yet comfortable with synthetic division, you can use polynomial division and the divisor will be x+4.


(Not showing the synthetic division steps or process)


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Trying to clear confusion:
Polynomial division works the same way as regular Long Division;
Account must be made for ALL terms of the powers of x, whether shown in the function or not; if not present in the function, then their coefficients are 0.
Remainder of zero means the value tested IS a root of the function;
<s>Remainder being non-zero means that the remainder is the value of the function at that quantity used as the "divisor" in synthetic division.  In other words, the possible root tested gives a remainder which is the value of the function at that possible root tested.</s>


The actual Remainder Theorem and Factor Theorem express that better.  This is in your College Algebra/Pre-Calculus textbook.


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The table of processing data for synthetic division:


_______-4_____|______4_____0_____-64_____0_____1_____0_____-18
______________|
______________|___________-16_____64______0_____0____-4_____16
______________|____________________________________________________
____________________4_____-16_____0_____0______1_____-4_____<b>-2</b>


The result for this specific example:
{{{highlight_green(highlight(f(-4)=-2))}}}