Factor Theorem

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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