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Mathematics

Quarter 2 – Week 1

Illustrating Polynomial Functions

What I Need to Know

In this module, you need to recall what you have learned about

polynomials like the degree, coefficients, constant terms, factoring, etc.

The module is divided into two lessons, namely:

 Lesson 1: Definition of Polynomial Function

 Lesson 2: Writing Polynomial Functions in Standard Form and in Factored

Form

After you go through this module, you are expected to:

1. illustrates polynomial functions
2. write polynomial function in standard form and in factored fo

Lesson

1

Definition of Polynomial

Functions

What I Need to Know

At the end of the lesson, you should be able to:

1. illustrates polynomial function;
2. identify polynomial function; and
3. determine the degree, the leading term and coefficient and the constant term.

What I Know

Directions: Choose the letter that best answers each question.

1. What function is y = x

3

• 2 x + 1

?

A. Linear Function B. Polynomial Function

C. Quadratic Function

B. Rational Function

2. What is the value of n in f

(

x

)

= x

n

if f is a polynomial function?

A.

√ 3

B.

C.

− 3

D.

3. Which of the following is NOT a polynomial function?

A.

P ( x )= ax + b

B.

P ( x )=

p ( x )

q ( x )

C. P

(

x

)

= a x

2

• bx + c

D.

P

(

x

)

= a x

4

• b x

3

• c x

4

• dx + e

4.. What is the leading coefficient of the function f(x) = 3x

3

+6x

2

• x/2 +2x

4

A. 0 B. 2 C. 4 D. 6

5. what type of polynomial function is P(x) = (x=2)(x-2)?

A. Cubic Polynomial Function

B. Quadratic Polynomial Function

C. Quartic polynomial Function

D. Zero Polynomial Function

What’s In

You have learned in the last module that to solve problems involving

polynomials, you must follow steps to have an easy solution.

Start this module by recalling your knowledge on the concept of polynomial

expressions.

 The word polynomial is derived from Greek words “poly” which means many

and “nominal” which means terms, so polynomial means many terms.

 Polynomials are composed of constants (numbers), variables (letters) and

exponents such as 2 in x

2

. The combination of numbers, variables and

exponents is called terms.

 Example: 2 x

3

• x

2

 There are three (3) terms in this expression: 2 x

3

, x

2

∧ 1

, where 1 is the constant,

x is the variable and 3 and 2 are the exponents.

This knowledge will help you understand the formal definition of polynomial

function.

What’s New

Let’s explore!

Directions: Complete the table below. State your reason if it is not a polynomial.

Expression Polynomial or Not Reason/s

10 x

x

2

3

• 3 x + 1

3. π

− 2 x

− 3

• x

3

1 − 4 x

2

Did you complete the table correctly? Do you remember when an expression

is a polynomial? A polynomial is an expression of one or more algebraic terms

each of which consists of a constant multiplied by one or more variable raised to a

nonnegative integral power.

What Is It

A polynomial function is a function of the form

P

(

x

)

= a

n

x

n

• a

n − 1

x

n − 1

• a

n − 2

x

n − 2

• … ++ a

1

x + a

0

,

a

n

≠ 0 ,

where

n is a nonnegative integer ,

a

0

, a

1

, … , a

n

are real numbers called coefficients

( numbers that appear in each term) , a

n

x

n

is the leading term ,

a

n

is the leading

coefficient , and

a

0

is the constant term (number without a variable). The highest

power of the variable of P ( x ) is known as its degree.

There are various types of polynomial functions based on the degree of the

polynomial. The most common types are:

 Zero Polynomial Function (degree 0): P

(

x

)

= a x

0

= a

 Linear Polynomial Function (degree 1): P

(

x

)

= a x

1

• b = ax + b

 Quadratic Polynomial Function (degree 2): P ( x )= a x

2

• bx + c

 Cubic Polynomial Function (degree 3): P

(

x

)

= a x

3

• b x

2

• cx + d

 Quartic Polynomial Function (degree 4): P

(

x

)

= a x

4

• b x

3

• c x

2

• dx + e

where

a , b , c , d ∧ e are constants.

Other than P ( x ), a polynomial function can be written in different ways,

like the following:

f ( x )= a

n

x

n

• a

n − 1

x

n − 1

• a

n − 2

x

n − 2

• … + a

1

x + a

0

,

y = a

n

x

n

• a

n − 1

x

n − 1

• a

n − 2

x

n − 2

• … + a

1

x + a

0

,

Example:

Degree of

the

Polynomial

Type of

Function

Leadin

g Term

Leading

Coefficient

Constant

Term

y = 8 x

4

− 4 x

3

• 2 x + 22

4 Quartic

8 x

4

y = 3 x

2

• 6 x

3

• 2 x

3 Cubic 6 x

3

What’s More

Let’s do this…

A. Directions: Complete the table below. If the given is a polynomial function, give

the degree, leading coefficient and its constant term. If it is not, then just give

the reason.

Polynomial

Function

or Not

Reason Degree

Leading

Coefficient

Constant

Term

1. f ( x )= 0

f

(

x

)

= x

3

4

• 2 x + 2

f ( x )=

√

3 x

y = 1 + 2 x + x

3

P ( x )=

x

− 1

What I Have Learned

A. Directions: Fill in the blank with the choices provided in the box.

A __________(1)__________ is a function which involves only

________(2)____________ integer powers or only positive integer exponents. The

_________(3)_______ of any polynomial is the highest power present in it. In the

____(4)_____ polynomial function y = 4 + 2 x + x

3

, __(5)_____ is the leading term, 4 is

the ___(6)_____, 1 is the ___(7)______, and ___(8)____ is the degree.

polynomial function cubic nonnegative

constant term leading coefficient degree

x

3

What I Can Do

Directions: Give two polynomial functions of different degree of polynomial.

Identify the degree of polynomial, the type of polynomial, the leading coefficient and

its constant term.

Polynomial Functions

Degree of

Polynomial

Type of

Polynomial

Leading

Coefficient

Constant

Term

Assessment

Directions: Choose the letter that best answers each question.

1. Which of the following is the term with number without variable?

A. constant term

B. degree

C. leading term

D. polynomial

2. What is the value of n in f

(

x

)

= x

n

if f is a polynomial function?

A.

√ 3

B. 3 C. − 3

D.

3. Which of the following is NOT a polynomial function?

A. P ( x )= ax + b

B.

P ( x )=

p ( x )

q ( x )

C.

P ( x )= a x

2

• bx + c

D. P ( x )= a x

4

• b x

3

• c x

4

• dx + e

4. What type of polynomial function is ( x

)

= 5 x

3

• x

2

?

A. Cubic Polynomial Function

B. Quadratic Polynomial Function

C. Quartic Polynomial Function

D. Zero Polynomial Function

5. What type of polynomial function is f ( x )=( x + 2 )( 2 x − 8 )?

A. Cubic Polynomial Function

B. Quadratic Polynomial Function

C. Quartic Polynomial Function

D. Zero Polynomial Function

Additional Activities

Directions: Give two examples for each type of polynomials. Identify

the degree of polynomial, the leading term and the constant term.

Lesson

2

Writing Polynomial Functions

in Standard Form and In

Factored Form

What I Need to Know

At the end of the lesson, you should be able to:

1. write polynomial functions in standard form; and
2. write polynomial functions in factored form

What I Know

Directions: Choose the letter that best answers each question.

1. What is the product of ( x + 2 )( x + 5 )?

A.

x

2

• 3 x + 10

B.

x

2

− 3 x + 10

C.

x

2

• 7 x + 10

D. x

2

• 3 x − 3

2. What term has the highest exponent in f

(

x

)

=− 2 x

4

• x

6

• 3 x + 1?

A.

− 2 x

4

B.

x

6

C. 3 x D. 1

3. What is the standard form of the polynomial function in number 2?

A. f ( x )= x

6

− 2 x

4

• 3 x + 1

B. f

(

x

)

= 1 + x

6

− 2 x

4

++ 3 x

C.

f

(

x

)

= x

6

− 2 x

4

• 1 + 3 x

D. f

(

x

)

=− 2 x

4

• 3 x + x

6

4. What is the standard form of y = 20 x + 14 x

2

• 2 x

3

?

A.

y = 2 x

3

• 20 x + 14 x

2

B.

y = 14 x

2

• 20 x + 2 x

3

C.

y = 2 x

3

• 14 x

2

• 20 x

D.

y = 14 x

2

• 2 x

3

• 20 x

5. What is the factored form of the polynomial function in number 4?

A.

y = 2 x ( x + 5 )( x − 2 )

B.

y = 2 x ( x + 5 )( x + 2 )

C. y = 5 x ( x + 2 )( x + 2 )

y = 2 x ( x + 5 )¿

What’s In

A polynomial function is a function of the form

P

(

x

)

= a

n

x

n

• a

n − 1

x

n − 1

• a

n − 2

x

n − 2

• … ++ a

1

x + a

0

,

a

n

≠ 0.

The terms of a polynomial may be written in any order. However, if

they are written in decreasing powers of x, then the polynomial function is in

standard form.

Before you proceed, try to recall the following.

Types of Special Products

1. Square of Binomial

This special product results into Perfect Square Trinomial (PST).

¿ ¿

Example: ¿

2. Product of Sum and Difference of Two Terms

This results to Difference of Two Squares.

(

a + b

) (

a − b

)

= a

2

− b

2

Example: ( x + 2 )( x − 2 )= x

2

− 4

3. Square of Trinomial

This would result to six (6) terms.

¿

Example:

¿

4. Product of Binomials

The result is a General Trinomial. F.O.I.L (First, Outer, Inner, Last) method is

usually used.

( a + b ) ( c + d )= ac +( bc + ad ) + bd

Example: ( x + 2

) (

x + 3

)

= x

2

+

(

2 x + 3 x

)

¿ x

2

• 5 x + 6

5. Product of Binomial and Trinomial

The result is a Sum or Difference of Two Cubes.

( a + b )( a

2

− ab + b

2

)= a

3

• b

3

( a − b ) ( a

2

• ab + b

2

)= a

3

− b

3

Example: ( x + 2 )( x

2

− 2 x + 4 )= x

3

Methods of Factoring

Method When is it

Possible

Example

1. Factoring

out the

Greatest

Common

Factor

(GCF)

If each term in

the polynomial

has a common

factor.

2 x

2

• 8 x

The common factor of both

terms is 2x.

2 x

2

• 8 x = 2 x ( x + 4 )

2. The Sum-

Product

Pattern (A-C

Method)

If the polynomial

is of the form

x

2

• bx + c

and

there are factors

of c that if added

will get

b .

x

2

• 5 x + 6

The factors of 6 that if added

will get 5 are 2 and 3.

x

2

• 5 x + 6 =

(

x + 2

) (

x + 3

)

3. Grouping

Method

If the polynomial

is of the form

ax

2

• bx + c

and

there are factors

of

ac that if

added will get b.

Steps:

 Split up middle

term.

 Group the terms.

 Factor out GCFs

of each group.

 Factor out the

common

binomial.

2 x

2

• 9 x − 5

The factors of ac =( 2 ) (− 5 )=− 10

that if added will get 9 are 10

and

− 1

.

 Split up middle term

2 x

2

• 9 x − 5 = 2 x

2

• 10 x − 1 x − 5

 Group the terms (make sure to

group the terms with common

factors)

¿( 2 x ¿¿ 2 − 1 x )+( 10 x − 5 )¿

 Factor out GCFs of each group

¿ x ( 2 x − 1 ) + 5 ( 2 x − 1 )

 Factor out the common binomial

¿ ( 2 x − 1 ) ( x + 5 )

4. Perfect

Square

Trinomials

If the first and

last terms are

perfect squares

4 x

2

• 12 x + 9

The first and last terms are

perfect squares:

and the middle

term is twice the

product of their

roots.

√

4 x

2

= 2 x

√

9 = 3

The middle term is twice the

product of their roots:

2 ( 2 x ) ( 3 )= 12 x

4 x

2

• 12 x + 9 =¿

5. Difference of

Squares

If the expression

represents a

difference of two

squares

x

2

− 4

Square roots of the terms:

√

x

2

= x

√

4 = 2

x

2

− 4 =( x + 2 )( x − 2 )

What’s New

Directions: Complete the table below.

Polynomial Function

Term

with

highes

t

expon

ent

Term/s with lower

exponents in

descending order

C

on

st

an

t

te

r

m

y =− 4 x

2

• x

4

− 45

y = 6 x

2

• 4 x + 3 x

3

What Is It

Writing Polynomial Function in Standard Form

When giving a final answer, you must write the polynomial function in

standard form. Standard form means that you write the terms by decreasing

exponents.

Here’s what to do:

1. Write the term with the highest exponent first.
2. Write the terms with lower exponents in descending order.
3. Remember that a variable with no exponent has an understood exponent of 1.
4. A constant term always comes last.

Examples: Write the following polynomial functions in standard form.

y = 1 + 2 x + x

5

− 4 x

3

• 2 x

4

• 5 x

2

Te

rm

wit

h

hig

he

st

ex

po

ne

nt

Term/s

with lower

exponents

in

descendin

g order

Co

nst

ant

ter

m

Standard form

x

5

2 x

4

,

− 4 x

3

,

5 x

2

,

2 x

y = x

5

• 2 x

4

− 4 x

3

• 5 x

2

• 2 x + 1

f

(

x

)

= 5 x + 9 x

2

− 3 x

8

Often, the polynomial function does not contain all of the exponents. You still

follow the same procedure listing the highest exponent first (8) then the next

(2) and finally the term with just a variable (understood exponent of 1).

Term

with

Term/s with

lower

Con

stan

Stan

highes

t

expon

ent

exponents

in

descending

order

t

ter

m

dard form

− 3 x

8

9 x

2

,

5 x

0

f

(

x

)

=− 3 x

8

• 9 x

2

• 5 x

3. y = x ( x

2

− 5 )

With a factored form of a polynomial function, you must find the product first.

In finding the product of a monomial and a binomial, recall the Distributive

Property.

Multiply the monomial to

the first term of the

binomial

x

(

x

2

)

= x

1 + 2

= x

3

Multiply the monomial to

the second term of the

binomial

x (− 5 )=− 5 x

Arrange the exponents in

descending order.

Therefore, the standard

form is

y = x

3

− 5 x

4. f ( x )=− x ( x − 4 )( x + 4 )

Use the special product, Sum and Difference of two terms, in answering this

function.

Get the product of the sum and

difference of two terms.

( x − 4 ) ( x + 4 )= x

2

− 16

Multiply -x to the product. − x ( x ¿¿ 2 − 16 )= x

3

• 16 x ¿

Thus, the polynomial

function in standard form

becomes

f ( x )= x

3

• 16 x.

Writing Polynomial Function in Factored Form

We will focus on polynomial functions of degree 3 and higher, since

linear and quadratic functions were already taught in previous grade levels.

The polynomial function must be completely factored.

Examples: Write the following polynomial functions in factored form.

y = 64 x

3

This is of the form a

3

• b

3

which is called the sum of cubes. The factored

form of a

3

• b

3

is ( a + b )( a

¿

¿ 2 − ab + b

2

). ¿ To factor the polynomial function follow the

steps below:

Find

a ∧ b

(

a is the cube root of the first term)

( b is the cube root of the second term)

a = 4 x

b = 5

Substitute the values of

a and

b in

( a + b )( a

¿

¿ 2 − ab + b

2

) ¿

y =( 4 x + 5 ) [( 4 x )¿¿ 2 −( 4 x ) ( 5 )+( 5 )

2

]¿

So the factored form is y =( 4 x + 5 ) ¿ ¿ )

y = 3 x

3

• 6 x

2

• 4 x + 8

This is of the form ax

3

• bx

2

• cx + d

. This can be easily factored if

a

b

=

c

d

. To

factor the polynomial function, follow the steps below:

Group the terms ( ax

¿

¿ 3 + bx

2

)+( cx + d )¿ y =( 3 x

¿

¿ 3 + 6 x

2

)+( 4 x + 8 )¿

Factor x

2

out of the first group of terms.

Factor the constants out of both groups.

y = x

2

( 3 x + 6 )+( 4 x + 8 )

y = 3 x

2

( x + 2 )+ 4 ( x + 2 )

Add the two terms by adding the

coefficients

y = 3 x

2

( x + 2 )+ 4 ( x + 2 )

y =( 3 x ¿ ¿ 2 + 4 )( x + 2 )¿

So, the factored form is y =( 3 x ¿ ¿ 2 + 4 )( x + 2 )¿

What’s More

A. Directions: Complete the table below.

Polynomial Function Ter

m

wit

h

hig

Term/s

with lower

exponents

in

descendin

C

o

n

st

a

St

a

n

d

ar

he

st

ex

po

ne

nt

g order

nt

te

r

m

d

fo

r

m

f

(

x

)

= 4 + 4 x

4

• 8 x

f ( x )=( x + 2 )( x − 2 )

B. Directions: Write the factored form of the following polynomial functions by

completing the table:

y = 343 x

3

Find

a ∧ b

( a is the cube root of the first term)

(

b is the cube root of the second

term)

a =

¿

¿

b =

¿

¿

Substitute the values of a and b in

( a + b )( a

¿

¿ 2 − ab + b

2

) ¿

y =¿ ¿

So, the factored form is

y =¿ ¿ )

y = 27 x

3

− 8

Find a ∧ b

( a is the cube root of the first term)

(

b is the cube root of the second

term)

a =

¿

¿

b =

¿

¿

Substitute the values of a and b in

( a − b )( a

¿

¿ 2 + ab + b

2

) ¿

y =¿ ¿

So, the factored form is y =¿ ¿)

What I Have Learned

A. Directions: Fill in the blanks with the correct word/s to complete each

statement.

_______(1)________ means that you write the terms by decreasing

exponents.

Steps in writing this form:

1. Write the term with the ____(2)_________ first.
2. Write the terms with lower exponents in ____(3)_________ order.
3. Remember that a variable with no exponent has an understood exponent of

(4).

4. A ______(5)_________ always comes last.

B. Direction: Factor the following:

y = x

4

− 512 x

y = 9 x

3

− 36 x

2

• 4 x − 16

What I Can Do

Directions: Write the standard form of the polynomial functions that is found in

nature.

1. The intensity of light emitted by a firefly can be determined by

L

(

t

)

= 10 +0.3 t +0.4 t

2

−0.01 t

3

.

2. The total number of hexagons in a honeycomb can be modeled by the

function f

(

r

)

= 1 + 3 r

2

− 3 r

.

Assessment

Directions: Choose the letter that best answers each question.

1. What is the product of ( x + 3 )( x + 3 )?

A. x

2

• 3 x + 9

B.

x

2

− 3 x + 9

C.

x

2

• 6 x + 9

D.

x

2

− 6 x + 9

2. What term has the highest exponent in f

(

x

)

= x

4

• 5 x

7

• 3 x

?

A. x

4

B. 5 x

7

C. 3 x D. 0

3. What is the standard form of the polynomial function in number 2?

A.

f

(

x

)

= 5 x

7

• x

4

• 3 x

B.

f

(

x

)

= 5 x

7

• 3 x + x

4

C. f

(

x

)

= x

4

• 5 x

7

• 3 x

D.

f

(

x

)

= 3 x + 5 x

7

• x

4

4. What is the factored form of the polynomial function y = 2 x

3

• 14 x

2

• 20 x

A. y = 2 x ( x + 5 )( x − 2 )

B. y = 2 x ( x + 5 )( x + 2 )

C.

y = 5 x ( x + 2 )( x + 2 )

D. y = 2 x ( x + 5 )( x + 5 )

5. How should f ( x )= x

3

− 64 be written in factored form?

A. f

(

x

)

=( x − 4 )( x

2

• 4 x + 16 )

B.

f

(

x

)

=( x + 4 )( x

2

• 4 x + 16 )

C.

f

(

x

)

=( x − 4 )( x

2

− 4 x + 16 )

D. f

(

x

)

=( x + 4 )( x

2

− 4 x + 16 )

Additional Activity

Directions: Give 3 situations where polynomial function is found and

write their standard form.

Summary

A polynomial function is a function of the form

P ( x )= a

n

x

n

• a

n − 1

x

n − 1

• a

n − 2

x

n − 2

• … ++ a

1

x + a

0

,

a

n

≠ 0 ,

where

n is a nonnegative integer ,

a

0

, a

1

, … , a

n

are real numbers called

coefficients (numbers that appear in each term) , a

n

x

n

is the leading term

(has the highest degree),

a

n

is the leading coefficient , and

a

0

is the constant

term (number without a variable). The highest power of the variable of

P ( x ) is

known as its degree.

When giving a final answer, you must write the polynomial function in

standard form. Standard form means that you write the terms by decreasing

exponents.

Here’s what to do:

1. Write the term with the highest exponent first.
2. Write the terms with lower exponents in descending order.
3. Remember that a variable with no exponent has an understood exponent of 1.
4. A constant term always comes last.

In writing polynomial function in Factored Form, make sure that it is

factored completely. The following questions might help you to factor the

polynomial functions completely.

1. Is there a common factor?
2. Is there a difference of squares?
3. Is there a perfect square trinomial?
4. Is there an expression of the form x

2

• bx + c

?

5. Are there factors of

ac that add up to

b ?

Assessment (Post-Test)

Directions: Choose the letter that best answers each question.

1. Which of the following is the value of

n in f ( x )= x

n

if

f is a polynomial function?

A.

√

2 B.

− 2

C.

2. Which of the following is NOT a polynomial function?

A.

f ( x )= 2021

B. f ( x )= 19

C.

f

(

x

)

= x

2

− x

f ( x )= √

3 x

2

3. What is the leading term of f

(

x

)

= x

2

• 4 x

3

?

A. x B. 2 C. 3

4 x

3

4. How should the polynomial function

f ( x )= x

4

− 8 x

2

+

x

+

• 4 x

3

be written in

standard form?

A. f

(

x

)

=− 8 x

2

+

• 4 x

3

• x

4

+

x

B.

f

(

x

)

=

x

− 8 x

2

+

• 4 x

3

• x

4

C.

f ( x )= x

4

• 4 x

3

− 8 x

2

+

x

+

D.

f ( x )= 4 x

3

+

− 8 x

2

+

x

• x

4

5. How should f

(

x

)

= x

4

• x

3

• x

2

• x be written in factored form?

A.

f

(

x

)

= x ( x + 1 )( x

2

+ 1 )

B. f ( x )= x ( 1 )( x

2

+ 1 )

C. f

(

x

)

= x ( x − 1 )( x

2

+ 1 )

D.

f ( x )= x (− 1 )¿ )