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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