Learning Guide Module
Subject Code Math 3 Mathematics 3
Module Code 4.0 Graphs of Polynomial and Rational Functions
Lesson Code 4.1 Remainder Theorem, Factor Theorem, and Descartes’ Rule of Signs
Time Frame 30 minutes
After completing this learning guide, you should be able to
1. find the remainder of a polynomial function using synthetic division,
long division, and remainder theorem;
2. identify if an expression is a factor of a polynomial function using
synthetic division, long division, and factor theorem;
3. identify the polynomial function from a given graph; and
4. enumerate the number of possible combinations of real and imaginary
zeros of a polynomial function using Descartes’ Rule of Signs.
With the COVID-19 crisis these past months, how did you overcome all the
challenges? How did your perception in life change? What type of ideas
and actions resulted to zero worries? Were they real or imaginary? Positive
or negative? Rational or Irrational?
In this section, we will determine the zeroes of the polynomial 10
which are the solutions of the equation 𝑓(𝑥) = 0 and each real MINUTES
zero is an 𝑥-intercept of the graph of 𝑓(𝑥).
TIP (The Important Point)
Fundamental Theorem of Algebra
If a polynomial 𝑓(𝑥) has a positive degree and complex coefficients, then 𝑓(𝑥) has
at least one complex zero.
Complete Factorization Theorem for Polynomials
If 𝑓(𝑥) is a polynomial degree 𝑛 > 0, then there exist n complex numbers
𝑐1 , 𝑐2 , . . . , 𝑐𝑛 such that 𝑓(𝑥) = 𝑎(𝑥 − 𝑐1 )(𝑥 − 𝑐2 ) ⋯ (𝑥 − 𝑐𝑛 ), where a is the leading
coefficient of 𝑓(𝑥). Each number 𝑐𝑘 is a zero of 𝑓(𝑥).
Mathematics 3|Page 1 of 9
, Remainder Theorem A polynomial function has a remainder 𝑓(𝑐)
when divided by 𝑥 − 𝑐
Example 1. Using synthetic division & remainder theorem
When 𝑓(𝑥) = 𝑥 3 + 9𝑥 2 + 17𝑥 + 5 is divided by 𝑥 + 3, what is the remainder?
Solutions.
By synthetic division,
−3ห 1 9 17 5
+ −3 −18 3
𝟏 6 -1 8
coefficients of quotient remainder
By remainder theorem,
𝑓(−3) = (−3)3 + 9(−3)2 + 17(−3) + 5
= −27 + 81 − 51 + 5
=8
Example 2. Using long division & remainder theorem
What is the remainder if 𝑓(𝑥) = 4𝑥 3 − 12𝑥 2 + 11𝑥 + 10 is divided by 2𝑥 − 5?
Solutions.
By long division,
2𝑥 2 − 𝑥 + 3 (quotient)
2𝑥 − 5 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
4𝑥 3 − 12𝑥 2 + 11𝑥 + 10
4𝑥 3 − 10𝑥 2
−2𝑥 2 + 11𝑥
−2𝑥 2 + 5𝑥
6𝑥 + 10
6𝑥 − 15
25 (remainder)
By remainder theorem,
5 5 3 5 2 5
𝑓 (2) = 4 (2) − 12 (2) + 11 (2) + 10
= 4(125
8
) − 12(25
4
) + 55
2
+ 10
= 25
Mathematics 3|Page 2 of 9
Subject Code Math 3 Mathematics 3
Module Code 4.0 Graphs of Polynomial and Rational Functions
Lesson Code 4.1 Remainder Theorem, Factor Theorem, and Descartes’ Rule of Signs
Time Frame 30 minutes
After completing this learning guide, you should be able to
1. find the remainder of a polynomial function using synthetic division,
long division, and remainder theorem;
2. identify if an expression is a factor of a polynomial function using
synthetic division, long division, and factor theorem;
3. identify the polynomial function from a given graph; and
4. enumerate the number of possible combinations of real and imaginary
zeros of a polynomial function using Descartes’ Rule of Signs.
With the COVID-19 crisis these past months, how did you overcome all the
challenges? How did your perception in life change? What type of ideas
and actions resulted to zero worries? Were they real or imaginary? Positive
or negative? Rational or Irrational?
In this section, we will determine the zeroes of the polynomial 10
which are the solutions of the equation 𝑓(𝑥) = 0 and each real MINUTES
zero is an 𝑥-intercept of the graph of 𝑓(𝑥).
TIP (The Important Point)
Fundamental Theorem of Algebra
If a polynomial 𝑓(𝑥) has a positive degree and complex coefficients, then 𝑓(𝑥) has
at least one complex zero.
Complete Factorization Theorem for Polynomials
If 𝑓(𝑥) is a polynomial degree 𝑛 > 0, then there exist n complex numbers
𝑐1 , 𝑐2 , . . . , 𝑐𝑛 such that 𝑓(𝑥) = 𝑎(𝑥 − 𝑐1 )(𝑥 − 𝑐2 ) ⋯ (𝑥 − 𝑐𝑛 ), where a is the leading
coefficient of 𝑓(𝑥). Each number 𝑐𝑘 is a zero of 𝑓(𝑥).
Mathematics 3|Page 1 of 9
, Remainder Theorem A polynomial function has a remainder 𝑓(𝑐)
when divided by 𝑥 − 𝑐
Example 1. Using synthetic division & remainder theorem
When 𝑓(𝑥) = 𝑥 3 + 9𝑥 2 + 17𝑥 + 5 is divided by 𝑥 + 3, what is the remainder?
Solutions.
By synthetic division,
−3ห 1 9 17 5
+ −3 −18 3
𝟏 6 -1 8
coefficients of quotient remainder
By remainder theorem,
𝑓(−3) = (−3)3 + 9(−3)2 + 17(−3) + 5
= −27 + 81 − 51 + 5
=8
Example 2. Using long division & remainder theorem
What is the remainder if 𝑓(𝑥) = 4𝑥 3 − 12𝑥 2 + 11𝑥 + 10 is divided by 2𝑥 − 5?
Solutions.
By long division,
2𝑥 2 − 𝑥 + 3 (quotient)
2𝑥 − 5 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
4𝑥 3 − 12𝑥 2 + 11𝑥 + 10
4𝑥 3 − 10𝑥 2
−2𝑥 2 + 11𝑥
−2𝑥 2 + 5𝑥
6𝑥 + 10
6𝑥 − 15
25 (remainder)
By remainder theorem,
5 5 3 5 2 5
𝑓 (2) = 4 (2) − 12 (2) + 11 (2) + 10
= 4(125
8
) − 12(25
4
) + 55
2
+ 10
= 25
Mathematics 3|Page 2 of 9
