Department of Education Mathematics 10 Polynomial Functions Second Quarter Week 1 Module

Department of Education

Mathematics 10

Polynomial Functions

Second Quarter – Week 1 Module

Schools Division Office – Muntinlupa City

Student Center for Life Skills Bldg., Centennial Ave., Brgy. Tunasan, Muntinlupa City (02) 8805-9935 / (02) 8805-9940

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EMELITA D. BAUTISTA EdD, Engr. ROLANDO S. MULDONG JOSEPH D. NILO, RANDY M. VARGAS

Quality Assurance Team Members

1

As you go through this module, you are expected to define and illustrate polynomial functions, identify the leading term, the leading coefficient, constant term and the degree of the polynomial function and be able to write the polynomial function in factored form.The ultimate goal of this module is for you to answer these questions: How are polynomial functions related to other fields of study? How are these used in solving real-life problems and in decision making?

Let us find out first what you already know related to the content of this module.

Answer all items. Choose the letter that best answers each question. Please take note of the items/questions that you will not be able to answer correctly and revisit them as you go through this module for self-assessment.

1. What should n be if f(x) = xⁿ defines a polynomial function?

A. An integer C. any number

B. A nonnegative integer D. any number except 0 2. Which of the following is an example of a polynomial function?

A. f(x)= ⁴

𝑥² + 3x -1 C. f(x)= √6 x – 2x⁴ B. f(x)= 2𝑥³² - ³

2x²

D. f(x)= x

+ 3x + 5

3. What is the leading coefficient of the polynomial function f(x) = 2x + x³ + 4?

A. 1 B. 2 C. 3 D. 4

4. How should the polynomial function f(x) = 2x+x³+3x⁵ + 4 be written in standard form?

A. f(x)=x³+2x+3x⁵+4 C. f(x)=4 + 2x + x³ + 3x⁵ B. f(x)= 4+3x⁵+2x+x³ D. f(x)= 3x⁵+x³+2x+4 5. If y=x(x+2)², the degree of the polynomial is

A. 1 B. 2 C. 3 D. 4 6. Express y= x² – 5x + 6 in factored form.

A. y = (x-2) (x-3) B. y = (x+2) (x-3) C. y = (x-2) (x+3) D. y = (x+2)(x+3)

2

7. Which of the following could be the value of n in the equation f(x) = xⁿ if f is a polynomial function?

A. -2 B. 0 C. ¹

4 D. √5

8. Given that f(x) = 7x^-3n + x², what value should be assigned to n to make f a function of degree 7?

A. - ⁷

3 B. - ³

7 C. ³

7 D. ⁷

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9. What is the constant term of the polynomial function f(x) = (x+5) (x-3) + 5 A. 10 B.-10 C. 5 D. -5

10. The leading term of y = (2x+3) (x+1) (x-2) (x+3) is

A. 2x⁴ B. 2x³ C. 2x² D. 2x 11. Using the given polynomial function in item #10, the constant term is A. 9 B. -9 C. 18 D. -18 12. The standard form of the polynomial function y= -x(x+3)(x-3) is

A. y = x² + 9x B. y = x³ + 9x C. y = -x² – 9x D. y = -x³ + 9x 13. In f(x) = x³ + x² + 18, the factored form of the polynomial function is

A. f(x) = (x²-2x+6) (x+3) C. f(x) = (x² + 2x +6) (x+3) B. f(x) = (x²+2x-6) (x+3) D. f(x) = (x²-2x+6) (x-3) 14. The constant term of the polynomial function f(x) = 2x³ + x² – 12x is A. 0 B. 1 C. 2 D. 3 15. Using the polynomial function in itam #14, what is the leading term?

A. 2x³ B x² C. -12x D. 0

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Build your understanding:

Which of the following algebraic expressions is a polynomial? If it is a polynomial, what is the degree?

6x + 1 -7x² + 5x + 3 ¹

4 x³-2x² -3x + 5 ⁴

𝑥² + 7x -3 ^{2𝑥2 − 1}

3𝑥+4 9√𝑥 + 3x -4 * The following expressions are polynomials: Why?

6x + 1 -7x² + 5x + 3 ¹

4 x³-2x² -3x + 5 The degree of 6x + 1 is 1, why?

The degree of -7x² + 5x + 3 is 2, why?

The degree of ¹

4 x³-2x² -3x + 5 is 3, why?

* The following expressions are not polynomials: Why?

⁴

𝑥² + 7x -3 ^{2𝑥2 − 1}

3𝑥+4 9√𝑥 + 3x -4

Did you answer each item correctly? Do you remember when an expression is a polynomial? Go back to the definition of polynomials in your previous lessons to be able to understand clearly why these expressions are not polynomials.

A polynomial function is a function of the form P(x) = anxⁿ +an-1x^n-1 + an-2x^n-2 + . . . +a1x + a0 , an ≠ 0

Where n is a nonnegative integer, a0, a1, . . . , an are real numbers called coefficients, anxⁿ is the leading term, an is the leading coefficient, and a0 is the constant term.

The terms of a polynomial may be written in any order. However, if they are written in decreasing powers of x, we say the polynomial function is in standard form.

Other than P(x), a polynomial function may also be denoted by f(x). Sometimes, a polynomial function is represented by a set P of ordered pairs(x,y). Thus, a polynomial function can be written in different ways, like the following:

f(x) = anxⁿ +an-1x^n-1 + an-2x^n-2 + . . . +a1x + a0 or

y = anxⁿ +an-1x^n-1 + an-2x^n-2 + . . . +a1x + a0

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Polynomials may also be written in factored form and as a product of irreducible factors, that is, a factor that can no longer be factored using coefficients that are real numbers like,

y = x⁴ + 2x³ –x² + 14x -56 in factored form is y = (x²+7) (x-2) (x+4) Examples:

1. Consider the given polynomial functions in the table below.

Polynomial Function Polynomial Function

in Standard Form Degree Leading

Coefficient Constant Term

1.f(x)=2-11x+2x² f(x)=3x² – 11x + 1 2 3 1

2.f(x)=2x³ + 5 + 15x f(x)=2x³ + 15x + 5 3 2 5

3.y=x(x²-5) y=x³ – 5x 3 1 0

4. y = 3x²-5x⁴ - 7 y = -5x⁴ + 3x² -7 4 -5 -7

5. y= -x(3x + 5) Y = -3x² – 15x 2 -3 0

To further illustrate the f(x) in no. 1 from the table. (Remember: in order the polynomial function to be in standard form, the terms should be written in decreasing powers of x.

Degree

f(x)=3x² – 11x + 1 Constant term

f(x)= 2x

+ 15x + 5

Leading coefficient

2. For each polynomial function, determine the degree of f(x), the leading coefficient and the constant term.

a. f(x) = (x-1)³(x-2)²(x-3) Solution:

f(x)= (x-1)(x-1)(x-1)(x-2)(x-2)(x-3)

When the factors are multiplied, the leading term of f(x) is x⁶ and the constant term is;

(-1)(-1)(-1)(-2)(-2)(-3) = 12

Hence, the degree of f(x) is 6, the leading coefficient is 1, and the constant term is 12.

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b. f(x)= (2x+1)²(x² – 2x +1) Solution:

f(x) = (2x+1)(2x+1)(x² -2x +1)

when the factors are multiplied, the leading coefficient is (2) (2)(1) = 4, the leading term is 4x⁴, and the constant term is (1)(1)(1)= 1.

Therefore, the degree of f(x) is 4, the leading coefficient is 4, and the constant term is 1.

3. Write the factored form of the following polynomial function, a. f(x)= x³ – x² – 4x + 4

Solution:

Using synthetic division, we can determine the zeros of f(x).

1 -1 -4 4 1

1 0 -4

1 0 -4 0 2

2 4

1 2 0 -2

-2

1 0

The zeros are 1 , 2 , and -2 therefore; x=1 , x=2 , x = -2 working backward;

x-1 , x-2 , x+2,

therefore, the factored form of the polynomial function is;

f(x) = (x-1)(x-2)(x+2) b. f(x) = x³ – 8x² + 15x

Solution:

Since x³ – 8x² + 15x has a common factor which is x, x³ – 8x² + 15x = x(x²- 8x + 15)

we can used the concept of factoring or synthetic division to get the factors of the trinomial x² – 8x + 15

the complete factors are (x-5)(x-3)

therefore, the factored form of the polynomial function is f(x) = x(x-5) (x-3)

c. f(x) = 2x⁴ – x³ – 14x² + 19x -6

using synthetic division, the zeros are 1, 2, -3 1nd ½ therefore, x = 1 , x=2 , x= -3 , x = ½

working backward, x-1 , x-2, x+3, x- ¹

2 or 2x -1 the factored form of f(x) = 2x⁴ – x³ – 14x² + 19x -6 is f(x) = (x-1)(x-2)(x+3)(2x-1)

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A. Determine whether the given function is a polynomial or not. If it is a polynomial function, determine its degree, leading coefficient, leading term, and constant term.

P(x) Yes/no Degree Leading

Coefficient Leading

Term Constant Term 1.f(x)= 2x⁴ – 9

2.f(x) = 3x^-2 + 4x^-3 – 5x 3.p(x) = (2x-3)²(3x+1) 4.y = 3x¹² - ²

x

5.f(x) = 12 – 4x⁵ + 3 6.y= 3x^-5 + 2x⁴ 7. y = -x(x+2)^1/2 8. y = √3 x – 5 9.y = √3𝑥 – 5

10.y = 4 + 2x² – 5x⁴+2x³

B. Give the degree, the leading term, and the constant term of each polynomial function.

Degree Leading

Term

Constant Term 1.f(x)= (x+1)(x-3)(x+3)

2.f(x) = (x+1)²(x-1)⁴ 3.y = (2x-3)(x-1) 4.y= (x+2)(x-3)(2x-1) 5.y = (3x+2)(x-2)³ 6. f(x) = x(3x+1) 7. f(x)= -x(-2x² -4) 8. y= (x+5)(x-5) 9.y= (x+1)⁵ (x+1)⁶ 10.f(x) = (2x-1)(x+1)

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A polynomial function of degree n is a function defined by

f(x)=anxⁿ+an-1x^n-1 + . . . + a2x²+a1x+a0, where a, an, an-2, . . . ,a1,,a0 are real numbers, an≠0, and n is a non-negative integer.

The leading term is anxⁿ, the leading coefficient is an, and the constant term is a0. The domain of a polynomial function is the set of all real numbers.

Give the degree, the leading term and the constant term of each polynomial function.

Degree Leading term Costant Term 1. f(x) = 2x⁴ + x³ – 2x²+3

2. f(x) = (x²-2x+6) (x-3) 3. y = (x²-5) (x-1) 4. y = 2x³ -3x⁴ -6x²+ 6x + 8

5. f(x) = (x³-1) (x+1) 6.f(x) = (3x-5) (x³ + 1)

7.y = – (x+3) (x+1)² (2x-5) 8. y = x²(x+3) (x-1)³ 9. y = (x+2)²(x-2)

10. f(x) = (x-1) (x+1) (x+1) 11. y = (2x+1) (x-3) 12. y = (2x+1) (x+3) (x+2)

13. y = (2x-1) (x+3) (x-2) (x1) (2x+1) 14. y = - {(x+3) (x-1)}

15. y = (x+2) (2x²+7x+3)

8

Read each item carefully. Write down the letter that corresponds to the correct answer.

1. Which of the following is NOT a polynomial expression?

A. x^-4 + 2x B. -2014x C. x² + 2x + 5 D. -5x³ + 2x – 7 2. Given f(x)= (x+1)² ( x-1)³, the degree of the polynomial function is

A. 2

B. 5 C. 1 D. 6

A. 2 B. 3 C. -1 D. 1 4. In f(x)= x³+x² + 18, the factored form is

A. (x²+2x+6)(x+3) C.(x²-2x+6)(x+3) B. (x²-2x+6)(x-3) D.(x²-2x-6)(x+3)

5. If the terms of a polynomial are written in decreasing powers of x, the polynomial function is in

A. Factored form C. Decreasing form

B.

General form D. standard form

6. What is the leading coefficient of the polynomial function f(x)= 3x + x² + 4?

A. 1 B. 2 C. 3 D. 4

7. How should the polynomial function f(x) = 3 + 2x³ – 5x + x² be written in standard form?

A. f(x) = 2x³ -5x + x²+ 3 C. f(x) = 3-5x+x²+2x³

B. f(x) = 2x³ + x² -5x +3 D. f(x)= -2x³ - x² +5x -3 8. A polynomial function whose degree is 0 is called

A. Linear function C. Constant function B. Quadratic Function D. Exponential function 9. The constant term in y= -x(x+3)(x-3) is

A. -1 B. 0 C. 3 D. 9 10. The factored form of f(x)= x³ + x² + 18 is

A. f(x) = (x²-2x+6)(x+3) C. f(x) = (x²+2x+6)(x+3) B. f(x) = (x²-2x+6)(x-3) D. f(x) = (x²-2x-6)(x+3) 11. In f(x) = (x²+5x+6)(x+1), the degree of the polynomial function is

A. 1 B. 2 C. 3 D. 4 12. Using the given polynomial function in item #11, the constant term is A. 1 B. 2 C. 5 D. 6 13. The factored form of f(x) = 4x³ + 16x² + 9x – 9 is

A. f(x) = (x+3)(2x+3)(2x-1) C. f(x) = (x+3)(2x+3)(2x-1) B. f(x) = (x+3)(2x-3)(2x-1) D. f(x) = (x-3)(2x+3)(2x-1) 14. The leading coefficient of the polynomial function in item #13 is A. 4 B. 16 C. 9 D. -9 15. If f(x) = 2(x² – 4x + 1), the constant term is

A. 1 B. 2 C. -4 D. 4

9 KEY TO CORRECTIONS

PRE-TEST POST-TEST

1. B 6. A 11. D 1. A 6. A 11. C

2. C 7. B 12. D 2. B 7. B 12. D

3. A 8. A 13. A 3. C 8. C 13. A

4. D 9. B 14. A 4. C 9. B 14. A

5. C 10. A 15. A 5. B 10. A 15. B

ACTIVITIES (A)

YES/NO Degree Leading

Coefficient Leading

Term Constant Term

1. Yes 4 2 2x⁴ -9

2. No

3. Yes 3 12 12x³ 0

4. No

5. yes 5 -4 -4x⁵ 15

6. No 7. No

8. Yes 1 √3 √3 x -5

9. No

10 yes 4 -5 -5x⁴ 4

ACTIVITIES (B)

Degree Leading Term Constant Term

1. 3 X³ -9

2. 6 X⁶ -64

3. 2 2x² 3

4. 3 2x³ 6

5. 4 27x³ -16

6. 2 3x² 0

7. 3 2x³ 0

8. 2 X² -25

9. 11 X¹¹ 1

10. 2 2x² -1

Check Your Understanding

Degree Leading Term Constant term

1. 4 2x⁴ 3

2. 3 x³ -18

3. 3 x³ 5

4. 4 -3x⁴ 8

5. 4 x⁴ -1

6. 4 3x⁴ -5

7. 4 -2x⁴ -15

8. 6 x⁶ -3

9. 3 x³ -8

10 3 x³ -1

11. 2 2x² -3

12. 3 2x³ 6

13. 5 4x⁵ -6

14. 2 -x² 3

15. 3 2x³ 6

References:

Learner’s Module for Mathematics (Grade 10), 2015, Department of Education,Phil.

Bernabe, Julieta G., et al. (2014), Our World of Math 10,Vibal Group,Inc.,Quezon City Concepcion, Benjamin, et.al. (2004), Algebra with Recreational Math,Ymas Publishing,Manila Oronce,Orlando A..et.al,(2003),Exploring Mathematics,Rex Bookstore,Inc. Manila