remainder theorem long division method

1. Polynomial Function
Remainder Theorem
Prepared by:
MARICEL T. MAS
T-1/ LIPAY HIGH SCHOOL

2. What do you think is/are the reason/s why
long division teaches in grade school ?

3. Long Division
Find 2359 ÷ 51 by hand
Which one goes inside
51 2359

4. Find 2359 ÷ 51 by hand
4
51 2359
204
319
- 306
13
-
6 13/51
Remainder

5. Lets do the same with a Polynomial
Divide x3 - 3x2 + x + 4 by x- 2
x- 2 √ x3 - 3x2 + x + 4

6. How do you divide a polynomial by
another polynomial?
• Perform long division, as you do with
numbers! Remember, division is repeated
subtraction, so each time you have a new
term, you must SUBTRACT it from the
previous term.
• Work from left to right, starting with the
highest degree term.
• Just as with numbers, there may be a
remainder left. The divisor may not go into
the dividend evenly.

7. Divide x3 - 3x2 + x + 4 by x- 2
x- 2 √ x3 - 3x2 + x + 4
x2 - 1
-x3 2x2
-x2 + x
x-
-x2
+ 2x
- x 4
Remainder
2
𝑥 − 2
+
-x + 2
2
+
-
-
-

8. What is Remainder Theorem?
• The remainder theorem states:
If the polynomial P(x) is divided by (x – c), then the
remainder is P(c).

9. Example:
Divide x3 - 3x2 + x + 4 by x- 2
Solution:
P(x)= x3 - 3x2 + x + 4
Since x-c = x- 2
c= 2
Therefore,
P(c) =P(2) = (2)3 – 3(2)2 + (2) + 4
= 8 – 12 + 2 + 4
= 2 remainder

10. Example 2. (by remainder theorem)
Divide x4 – 3x3 + 2x – 2 by x+ 2
Solution:
P(x) = x4 – 3x3 + 2x – 2
x – c = x+ 2
c = -2
P(c) = P(-2) = (-2)4 – 3(-2)3 + 2(-2) – 2
= 16 + 24 – 4- 2
= 34 remainder

11. To check by using long division
Divide x4 – 3x3 + 2x – 2 by x+ 2
x+2 x4 – 3x3 + 2x – 2
x3
x4 + 2x3
-5x3
--5x3 10x2
-5x2
-
10x2
+ 2x
+ 10x
10x2 + 20x
-
-18x - 2
-18
-18x - 36
34
-
Remainder

12. Now its your turn:
• Find the remainder of the following :
• a3 – 3a2 - a + 20; a+ 2
• x3 + 14x2 +47x – 1; x + 7
• 2x3 – 8x2 + 19x– 12; x- 3

13. Evaluation
Find the remainder of the following using the
remainder theorem:
• 2x3 – 15x2 + 11x + 10 ; x – 5
• 2a3 – 13a2 – 20a + 25; a+3
• 2y3 -5y2 – 8y – 50; y-5

14. Assignment:
Use the remainder theorem to find the remainder of
the following function:
• x3 – 4x2 + x + 6; x-2
• x3 – 2x2 – 75; x – 5
• x3 – 11x – 6; x + 3