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) About Me  (Show Source):
You can put this solution on YOUR website!
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%28x%29=4x%5E6%2B0x%5E5-64x%5E4%2B0x%5E3%2Bx%5E2%2B0%2Ax-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;
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.

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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_____-2

The result for this specific example:
highlight_green%28highlight%28f%28-4%29=-2%29%29

Answer by MathTherapy(10552) About Me  (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. 

f%28x%29+=+4x%5E6+-+64x%5E4+%2B+x%5E2+-+18

From remainder theorem, f%28-+4%29+=+4%28-+4%29%5E6+-+64%28-+4%29%5E4+%2B+%28-+4%29%5E2+-+18 ----- Substituting - 4 for x to determine remainder

f%28-+4%29+=+4%284096%29+-+64%28256%29+%2B+16+-+18

f(- 4) = 16,384 – 16,384 + 16 - 18

f(- 4), or remainder is: highlight_green%28-+2%29

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.