math 4 topics

1. FINDING VALUES OF
POLYNOMIAL FUNCTIONS
- SYNTHETIC DIVISION
REYNALDO B. PANTINO
Teacher II

2. OBJECTIVES:
1.) To determine the
remainder of
polynomials using;
a.) Synthetic Division
2.) To demonstrate
understanding of
remainder theorem
versus synthetic division

3. Recall:
1. If a polynomial P(x)
is divided by x – c,
where c is a real
number, then P(c) is
what we call?
2.) If x – c, what makes
x equals?
3.) What is Division
Algorithm for
Polynomials?
4.) P(x) stands for ____.
5.) Q(x) stands for ___.
6.) x – c stands for ___.
7.) P(c) stand for ___.
8.) Is P(c) equals R?
9.) R stands for _____.

4. Finding the values of polynomial functions
Synthetic Division VS Remainder Theorem
Illustrative Example 1.
A. Use synthetic division and the Remainder theorem to find
the value of P(x) = 2 x3 -8 x2 +1 9 x -12 at 3
.
Solution: (By synthetic division)
3 2 -8 19 -12
6 -6 39
2 -2 13 27
Remainder 27

5. Finding the values of polynomial functions
Synthetic Division VS Remainder Theorem
Solution:(By Remainder Theorem)
P(x) = 2x3 – 8x2 + 19x – 12
P(3) = 2(3)3 – 8(3)2 + 19(3) – 12
P(3) = 54 – 72 + 57 – 12
P(3) = 27
The remainder is 27. Therefore, P(3) = 27

6. Finding the values of polynomial functions
Synthetic Division VS Remainder Theorem
Illustrative Example 2.
Divide P(x) = ax2 + bx + c by x – c.
Solution: (by synthetic division)
c a b c
ac ac2 + bc
a ac + b ac2 + bc + c
The remainder is

7. Finding the values of polynomial functions
Synthetic Division VS Remainder Theorem
Solution:(By Remainder Theorem)
P(x) = ax2 + bx + c ( x – c )
divisor
P(c) = a(c)2 + b(c) + c
P(c) = ac2 + bc + c
This shows that the remainder is indeed
equal to P(c).

8. Finding the values of polynomial functions
Synthetic Division VS Remainder Theorem
Illustrative Example 3.
Use the synthetic division and the remainder theorem to
find the value of P(a) = 3a3 + 13a2 + 9a – 15 at a = - 4
Solution by synthetic division
-4 3 13 9 -15
3
-12
1
-4
5
-20
-35

9. Finding the values of polynomial functions
Synthetic Division VS Remainder Theorem
Solution:(By Remainder Theorem)
P(a) = 3a3 + 13a2 + 9a – 15 at a = - 4
P(-4) = 3(-4)3 + 13(-4)2 + 9(-4) – 15
P(-4) = -192 + 208 – 36 – 15
P(-4) = -35
The remainder is -35. Therefore, P(-4) = -35

10. Finding the values of polynomial functions
Synthetic Division VS Remainder Theorem
Exercises. Find the value of the polynomial function using
synthetic division and the remainder theorem.
1.) P(x) = x3 – 4x2 + x + 6 x = 2
2.) P(x) = x3 – 2x2 – 75 x = 5
3.) P(x) = x3 – 11x – 6 x = -3
4.) P(x) = x3 + 8x2 + 13x – 10 x = -5
5.) P(x) = 2x3 – 9x – 5x2 – 8 x = 4
6.) P(x) = 8x + x3 – 5x2 + 2 x = 2

11. In summary, the remainder R obtained in
synthetic division of f(x) by x – c, provides these
information:
1.) The remainder R gives the value of f
at x =c, that is R = f(c).
2.) If R = 0, then x – c is a factor of f(x).
3.) If R = 0, then (c, 0) is an x intercept
of the graph of f.

12. ASSIGNMENTS:
What is factor theorem?
Copy the illustrative examples in
your notebook.
Reference: Advanced Algebra,
Trigonometry & Statistics.
pp. 100 - 101