Remainder theorem, Factor theorem, and Rational root theorem

Hi everyone, it's Amina and this blog is about our lesson on remainder theorem, factor theorem, and rational root theorem.

Remainder Theorem:
The Remainder Theorem is useful for evaluating polynomials at a given value of x.
The Remainder Theorem starts with an unnamed polynomial p(x), where "p(x)" that means "some polynomial p whose variable is x". Then dividing that polynomial by a linear factor x – a, where a is just some number. As a result of the synthetic division, you end up with some polynomial answer q(x) (the "q" standing for "the quotient polynomial") and some polynomial remainder r(x).

Example:
p(x) = x³ – 7x – 6 and divide be x-4( so a =4)  




                         
 We get a quotient of q(x) = x² + 4x + 9 , with a remainder of r(x) = 30.

If   ^p(x) / _{(x – a)}  =  q(x)  with remainder  r(x),
then  p(x)  =  (x – a) q(x)  +  r(x).

Since  ^{(x^3 – 7x – 6)} / _{(x – 4)}   =   x² + 4x + 9   with remainder 30,   
 then   x³ – 7x – 6  =   (x – 4) (x² + 4x + 9)  +  30.

Another way to get the remainder  is we replace x with 4.
 p(4) = (4 – 4)((4)² + 4(4) + 9) + 30        = (0)(16 + 16 + 9) + 30        = 0 + 30        = 30

Factor Theorem: 

 The factor theorem is a theorem commonly applied to factorizing and finding the roots of polynomial equations. If P(x), a polynomial in x is divided by x - a and the remainder is zero, then (x - a) is a factor of P(x)

Example:
Is x - 2 is a factor of x² - 7x + 10.

Let P(x) = x² - 7x + 10
By the factor theorem, x - 2 is the factor of P(x) if P(2) = 0.

P(x) = x² - 7x + 10
P(2) = 2² - 7 * 2 + 10 (Substitute x = 2)
= 4 - 14 + 10
= 0
Since P(2) = 0  => x - 2 is a factor of P(x).

Rational Root Theorem:

Steps to solving rational root theorem questions:

1. Arrange the polynomial in descending order
2. Write down all the factors of the constant term. These are all the possible values of p .
3. Write down all the factors of the leading coefficient. These are all the possible values of q .
4. Write down all the possible values of . Remember that since factors can be negative, and - must both be included. Simplify each value and cross out any duplicates.

     5.Use synthetic division to determine the values of for which P() = 0 . These are all the rational roots of P(x) . 

Example:

1. P(x) = 2x ⁴ + x ³ -19x ² - 9x + 9
2. Factors of constant term: ±1 , ±3 , ±9 .
3. Factors of leading coefficient: ±1 , ±2 .
4. Possible values of : ± , ± , ± , ± , ± , ± . These can be simplified to: ±1 , ± , ±3 , ± , ±9 , ± .
5. Use synthetic division:
   

Thus, the rational roots of P(x) are x = - 3 , -1 , , and 3 .

Sources:
 http://www.purplemath.com/modules/remaindr.htm
 http://math.tutorvista.com/algebra/factor-theorem.html
 http://www.sparknotes.com/math/algebra2/polynomials/section4.rhtml