We will discuss here about the basic concept of Factor Theorem.
If the polynomial p(x) is divided by x  α then by division algorithm,
P(x) = (x  α) q(x) + R,
where q(x) is the quotient and R is the remainder which is a constant.
Putting x = α in P(x) = (x  α) q(x) + R, we get,
p(α) = (α  α) q(α) + R
⟹ p(α) = R.
If R = p(α) = 0, then
p(x) = (x  α) q(x) and so (x  α) is a factor of p(x).
x  α is a factor of p(x) if p(α) = 0, and if p(α) = 0 then p(x) has a factor x  α.
Example on Factor Theorem:
Prove that x + 5 is a factor of 2x\(^{2}\) + 7x  15. Now, x + 5 = x – (5) and
p(5) = 2 (5)\(^{2}\) + 7(5)  15
= 50  35  15
= 0.
So, x  (5), i.e., x + 5 is a factor of 2x\(^{2}\) + 7x  15
Corollary: ax + b is a factor of p(x) if p(\(\frac{b}{a}\)) = 0, and if p (\(\frac{b}{a}\)) = 0 then p(x) has a factor x  α.
● Factorization
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