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
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)