Answer:
Step-by-step explanation:
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The Factor Theorem
The Factor Theorem is a result of the Remainder Theorem, and is based on the same reasoning. If you haven't read the lesson on the Remainder Theorem, review that topic first, and then return here.
As the Remainder Theorem points out, if you divide a polynomial p(x) by a factor x – a of that polynomial, then you will get a zero remainder. Let's look again at that Division Algorithm expression of the polynomial:
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p(x) = (x – a)q(x) + r(x)
If x – a is indeed a factor of p(x), then the remainder after division by x – a will be zero. That is:
p(x) = (x – a)q(x)
In terms of the Remainder Theorem, this means that, if x – a is a factor of p(x), then the remainder, when we do synthetic division by
x = a, will be zero.
The point of the Factor Theorem is the reverse of the Remainder Theorem: If you synthetic-divide a polynomial by x = a and get a zero remainder, then, not only is x = a a zero of the polynomial (courtesy of the Remainder Theorem), but x – a is also a factor of the polynomial (courtesy of the Factor Theorem).
Just as with the Remainder Theorem, the point here is not to do the long division of a given polynomial by a given factor. This Theorem isn't repeating what you already know, but is instead trying to make your life simpler. When faced with a Factor Theorem exercise, you will apply synthetic division and then check for a zero remainder.
Use the Factor Theorem to determine whether x – 1 is a factor of
f (x) = 2x4 + 3x2 – 5x + 7.
For x – 1 to be a factor of f (x) = 2x4 + 3x2 – 5x + 7, the Factor Theorem says that x = 1 must be a zero of f (x). To test whether x – 1 is a factor, I will first set x – 1 equal to zero and solve to find the proposed zero, x = 1. Then I will use synthetic division to divide f (x) by x = 1. Since there is no cubed term, I will be careful to remember to insert a "0" into the first line of the synthetic division to represent the omitted power of x in 2x4 + 3x2 – 5x + 7:
