Topics in Mathematics 4

1. The
Remainder
Theorem
REYNALDO B. PANTINO

2. Obj ect i ves
Define Division Algorithm for Polynomials
Define Remainder Theorem
Show the proof of the remainder theorem
Determine the remainder and quotient of
polynomials using;
a.) Division Algorithm for Polynomials
b.) Remainder Theorem

3. Think of this:
1. What is the largest positive integer less than
50 which has a remainder of 1 when divided by
2?
2. What is the positive integer which has a
remainder of 2 when divided by 3?
3. What is the positive integer which has a
remainder of 3 when divided by 5?
4. What is the positive integer which has a
remainder of 5 when divided by 7?

6. Consi der t hi s di vi si on:

9. Definition
Division Algorithm for Polynomials
For each polynomial P(x) of positive
degree n and any real number c, there exist a
unique polynomial Q(x) and a real number R
such that;
P(x) = (x – c) ● Q(x) + R
where Q(x) is of degree n – 1 and R is the
remainder

10. Definition
Remainder Theorem
If a polynomial P(x) is divided by x – c,
where c is a real number, then the remainder is
P(c).
Proof:
P(x) = (x – c) ● Q(x) + R
P(c) = (c – c ) ● Q(c) + R
P(c) = 0 ● Q(c) + R
P(c) = R
Hence, the remainder R is equal to P(c).

11. Illustrative Examples:
A.) Apply the Remainder theorem to find the remainder;
(x3 – 3x2 + x + 4) ÷ (x – 2)
Solution:
P(x) = x3 – 3x2 + x + 4
x – c = x – 2 then c = 2
Therefore;
P(2) = (2)3 – 3(2)2 + (2) + 4
P(2) = 2 r emainder

12. Illustrative Examples:
B.) Use the Remainder Theorem to find the remainder when;
(x4 – 3x3 + 2x – 2) ÷ (x + 2)
Solution:
P(x) = x4 – 3x3 + x – 2
c = – 2
Therefore;
P(-2) = (-2)4 – 3(-2)3 + 2(-2) – 2
P(-2) = 34 r e mainder

13. 10X2
5X3
5X3
10X2 2X
20X
18X

14. 18
10X2 2X
20X
2
36
18X
18X
34
Continuati
on
of the
solution
remainder

15. Exercises :
Find the remainder when the first polynomial is divided by the
second polynomial. Use the remainder theorem.
a3 – 3a2 – a + 20 a + 2
x3 + 14x2 + 47x – 12 x + 7
2x3 – 15x2 + 11x + 10 x – 5
2a3 – 13a2 – 20a + 25 a + 3
2y3 – 5y2 – 8y – 50 y – 5
3y3 + 2y2 – y + 5 y + 2

16. Assignments:
Finding Values of Polynomial
function using;
a.Synthetic Division
b.Remainder Theorem
Reference: Advanced Algebra, Trigonometry
& Statistics pp. 100 - 101