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

• Polynomial
• Polynomial Equation and its Roots
  
• Division Algorithm
  
• Remainder Theorem
  
• Problems on Remainder Theorem
  
• Factors of a Polynomial
  
• Worksheet on Remainder Theorem
  
• Factor Theorem
  
• Application of Factor Theorem

10th Grade Math

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