write factor theorem and give its proof also
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Proof of factor theorem: Consider a polynomial f (x) which is divided by (x – c) . Thus, by the Remainder theorem, Thus, (x – c) is a factor of the polynomial f (x).
Answer:
f(x) = (x – a)q(x) + r(x)
Step-by-step explanation:
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Proof of factor theorem:
Consider a polynomial f (x) which is divided by (x – c) .
Then, f (c) = 0.
Thus, by the Remainder theorem,
Thus, (x – c) is a factor of the polynomial f (x).
Proof of the converse part:
By the Remainder theorem,
f (x) = (x – c) q(x) + f (c)
If (x – c) is a factor of f (x), then the remainder must be zero.
That is, (x – c) exactly divides f (x).
Thus, f (c) = 0.
Hence proved.
Note:
The Remainder theorem says, if (x - c) divides the polynomial f (x), then the remainder is f (c) That is,
f (x) = (x – c) q(x) + f (c)
Suppose the remainder f (c) = 0, f (x) = (x – c) q(x).
Thus, (x – c) is the factor of f (x). Hence, it can be concluded that the “Factor theorem” is the reverse of “Remainder theorem”.
Example:
Consider a polynomial . Determine whether (x+1) is a factor of f (x).
By the Factor theorem, (x + 1) is a factor of f (x) if f (–1) = 0.
Obtain the value of f (–1).
Since f (–1) = 0, (x + 1) is a factor of f (x).