A few notes about the Remainder Theorem, examples of use and the factorization of polynomials

1. Remainder theorem and
factorization of polynomials
by Suso

2. Remainder theorem
The remainder of the division of a
polynomial P(x) by (x-a) is P(a).
For example, if P(x)=3x3-2x+5, the remainder
of the division of P(x) by (x-2) is equal to
P(2)=3·23-2·2+5=25

3. Exercise
What is the remainder of the division of
P(x)=5x4-3x3+3x2+x-2 by (x-3)?
Check your result using the synthetic division

4. Exercise
What is the remainder of the division of
P(x)=-4x3-4x2+9x by (x+2)?
Check your result using the synthetic division

5. Exercise
Is the division (8x3-4x+12)(x+1) exact?

6. Exercise
Is the division (2x5+3x4+2x3-4x2+12)(x-2)
exact?

7. Exercise
What is the remainder of the division
(2x35+3x24+12)(x+1)?

8. Root of a polynomial
A number a is called a root (or a zero) of a
polynomial P(x) if P(a)=0.

9. Root of a polynomial. Example
Let´s consider the polynomial
P(x)=x3-3x2-2x+6.
If we plug in x=3 into the polynomial, we get:
P(3)=33-3·32-2·3+6 ⇒ P(3)=0
Then, 3 is a root of the polynomial P(x)

10. Root of a polynomial. Example
Note that x-3 is a factor of the polynomial
P(x)=x3-3x2-2x+6.
(the division is exact)

11. Roots and factors of a polynomial
a is a root of a polynomial P(x)
↕
P(a)=0
↕
(x-a) is a factor of P(x)

12. Exercise
Is− 2 a root of P(x)=x3-3x2-2x+6?

13. Exercise
In that case, ...............is a factor of P(x).