If (x-3) is a factor of p(x), then the reminder is
Answers
Step-by-step explanation:
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:
completed division: 2 2 5 0 7
Since the remainder is not zero, then the Factor Theorem says that:
x – 1 is not a factor of f (x).
I think this would help you buddy ......,
