1. F a C Ro
t
Theorem
R
e
M
a
i
N
d
e
R
THEOREM

2. REMAINDER THEOREM: If
the polynomial P(X) is
divided by X-C, then
the remainder is P(C).
FACTOR THEOREM: If the
remainder comes out to
be 0 (zero), then X-C
is a factor of P(X).

3. E SM
a
x L
p e

4. f(x)= x4 – 13x2 + 36
𝑝
𝑞
=
±1,±2,±3,±4,±6,±9,±12,±18,±36
±1
Possible roots:
±1, ±2, ±3, ±4, ±6, ±9, ±12,
±18, ±36
1.

5. If x= 1:
f(1) = (1)4 – 13(1)2 + 36
= 1 – 13(1) + 36
= 1 – 13 + 36
= 24
Therefore, x=1 is not a root and
(x-1) is not a factor.
If x= -1:
f(-1) = (-1)4 – 13(-1)2 + 36
= 1 – 13(1) + 36
= 1 – 13 + 36
= 24
Therefore, x=-1 is not a root and
(x+1) is not a factor.

6. If x= 2:
f(2) = (2)4 – 13(2)2 + 36
= 16 – 13(4) + 36
= 16 – 52 + 36
= 0
Therefore, x=2 is a root and (x-2)
is a factor.
If x= -2:
f(-2) = (-2)4 – 13(-2)2 + 36
= 16 – 13(4) + 36
= 16 – 52 + 36
= 0
Therefore, x=-2 is a root and
(x+2) is a factor.

7. If x= 3:
f(3) = (3)4 – 13(3)2 + 36
= 81 – 13(9) + 36
= 81 – 117 + 36
= 0
Therefore, x=3 is a root and
(x-3) is a factor.
If x= -3:
f(-3) = (-3)4 – 13(-3)2 + 36
= 81 – 13(9) + 36
= 81 – 117 + 36
= 0
Therefore, x=3 is a root and
(x-3) is a factor.

8. Since the exponent of the
polynomial function is 4,
there should be four roots
and four factors.
The roots of the polynomial
function, f(x) = x4 – 13x2
+ 36, are ± 2 𝑎𝑛𝑑 ± 3 . The
factors of the polynomial
function are (x-2) (x+2)
(x-3) (x+3).

9. f(x) = x2– 12x + 9
𝑝
𝑞
=
±1,±3,±9,
±1
if x = 1
= 12- 10(1) + 9
= 1 -10 + 9
=0
Therefore, x=1 is a root and
(x-1) is a factor.
2.

10. If x = -1
= 12- 10(-1) + 9
= 1 +10 + 9
=20
Therefore, x=-1 is a not a
root and (x+1) is not a
factor.
if x = 3
= 32- 10(3) + 9
= 9 - 30 + 9
= -2
Therefore, x=3 is a not root
and (x-3) is not a factor.

11. if x = -3
= -32- 10(-3) + 9
= 9 + 30 + 9
= 48
Therefore, x=-3 is not a root
and (x+3) is a not factor.
if x = 9
= 92- 10(9) + 9
= 81 -90 + 9
= 0
Therefore, x=9 is a root and
(x-9) is a factor.

12. If x =- 9
= 92- 10(9) + 9
= -81 + 90 + 9
= 18
Therefore, x=-9 is not a root
and (x+9) is not a factor.
Since the exponent of the polynomial
function is 2, there should be two
roots and two factors.
The roots of the polynomial function,
f(x) = x2– 12x + 9 ate
+1 𝑎𝑛𝑑 + 9. The factors of the
polynomial function are (x-1) (x-9)

13. Danielle Erika L.
Diva
Ronalie C. Mejos
Mark Lenon F. Dacir
Rafael C. Vallejos
MEMBERS
X- Einstein