SOLUTION: Please help!! I don't know what I am doing!! Use the remainder theorem to find the remainder when f(x) is divided by x+4. Then use the factor theorem to determine whether x+4 is

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Question 970032: Please help!! I don't know what I am doing!!
Use the remainder theorem to find the remainder when f(x) is divided by x+4. Then use the factor theorem to determine whether x+4 is a factor of f(x).
f(x)=4x^6-64x^4+x^2-18

Thank you in advance!!
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
This mostly means the use of synthetic division to check if 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 .

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)

---
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;
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.

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

----
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_____-2

The result for this specific example:
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

Please help!! I don't know what I am doing!!
Use the remainder theorem to find the remainder when f(x) is divided by x+4. Then use the factor theorem to determine whether x+4 is a factor of f(x).
f(x)=4x^6-64x^4+x^2-18

Thank you in advance!!
Divisor of polynomial: x + 4, so x = - 4. 



From remainder theorem,  ----- Substituting - 4 for x to determine remainder



f(- 4) = 16,384 – 16,384 + 16 - 18

f(- 4), or remainder is: 

Since there's a remainder of - 2 when x + 4 is used as a factor, or when x = - 4, then x + 4 is NOT a factor.

Remainder should be 0 (zero) for a polynomial to be considered a factor of another polynomial. 

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