Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

quarter 1 grade 10 mathematics reviewer covering quater 1 of grade 10 mathematics

Typology: Summaries

2022/2023

1 / 21

Download grade 10 mathematics reviewer and more Summaries Mathematics in PDF only on Docsity! Mathematics Quarter 2 – Week 1 Illustrating Polynomial Functions What I Need to Know 1 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. 2 What Is It A polynomial function is a function of the form P ( x )=an x n+an−1 x n−1+an−2 x n−2+…++a1 x+a0, an≠0 , where n is a nonnegative integer , a0, a1 ,…,an are real numbers called coefficients (numbers that appear in each term) , an xn is the leading term, an is the leading coefficient, and a0 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 x0=a Linear Polynomial Function (degree 1): P ( x )=a x1+b=ax+b Quadratic Polynomial Function (degree 2): P ( x )=a x2+bx+c Cubic Polynomial Function (degree 3): P ( x )=a x3+b x2+cx+d Quartic Polynomial Function (degree 4): P ( x )=a x4+b x3+c x2+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 )=an x n+an−1 x n−1+an−2 x n−2+…+a1 x+a0, y=an x n+an−1 x n−1+an−2 x n−2+…+a1 x+a0, Example: Degree of the Polynomial Type of Function Leadin g Term Leading Coefficient Constant Term 1. y=8x4−4 x3+2 x+22 4 Quartic 8 x4 8 22 2. y=3 x2+6 x3+2x 3 Cubic 6 x3 6 0 5 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 2. f ( x )=x 3 4 +2x+2 3. f ( x )= 3 √3 x 4. y=1+2 x+x3 5. P ( x )= 3 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+2x+x3, __(5)_____ is the leading term, 4 is the ___(6)_____, 1 is the ___(7)______, and ___(8)____ is the degree. 6 polynomial function cubic nonnegative constant term leading coefficient degree 3 1 x3 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 1. 2. 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 )=xn if f is a polynomial function? A. √3 B. 3 C. −3 D. 1 3 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 x2+bx+c D. P ( x )=a x4+b x3+c x4+dx+e 4. What type of polynomial function is ( x )=5 x3+ x2+15 ? A. Cubic Polynomial Function B. Quadratic Polynomial Function 7 What’s In A polynomial function is a function of the form P ( x )=an x n+an−1 x n−1+an−2x n−2+…++a1 x+a0, an≠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 )=a2−b2 Example: (x+2)( x−2)=x2−4 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 )=x2+ (2 x+3 x )+6 ¿ x2+5x+6 5. Product of Binomial and Trinomial The result is a Sum or Difference of Two Cubes. (a+b)(a2−ab+b2)=a3+b3 (a−b ) (a2+ab+b2 )=a3−b3 Example: (x+2)( x2−2x+4 )=x3+8 10 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 x2+8 x The common factor of both terms is 2x. 2 x2+8 x=2 x (x+4) 2. The Sum- Product Pattern (A-C Method) If the polynomial is of the form x2+bx+cand there are factors of c that if added will get b. x2+5x+6 The factors of 6 that if added will get 5 are 2 and 3. x2+5x+6=( x+2 ) ( x+3 ) 3. Grouping Method If the polynomial is of the form ax2+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 x2+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 x2+9 x−5=2 x2+10 x−1 x−5 Group the terms (make sure to group the terms with common factors) ¿(2x¿¿2−1 x)+(10x−5)¿ Factor out GCFs of each group ¿ x (2 x−1 )+5 (2x−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 x2+12 x+9 The first and last terms are perfect squares: 11 and the middle term is twice the product of their roots. √4 x2=2 x √9=3 The middle term is twice the product of their roots: 2 (2x ) (3 )=12 x 4 x2+12 x+9=¿ 5. Difference of Squares If the expression represents a difference of two squares x2−4 Square roots of the terms: √ x2=x√4=2 x2−4=(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 1. y=−4 x2+x4−45 2. y=6x2+4 x+3 x3 What Is It Writing Polynomial Function in Standard Form 12 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. 1. y=64 x3+125 This is of the form a3+b3 which is called the sum of cubes. The factored form of a3+b3 is (a+b)(a¿¿2−ab+b2) .¿ 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+b2)¿ y= (4 x+5 ) [(4 x)¿¿2−(4 x )(5)+(5 )2]¿ So the factored form is y= (4 x+5 ) ¿¿) 2. y=3 x3+6 x2+4 x+8 This is of the form ax3+bx2+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+bx2)+(cx+d)¿ y=(3 x¿¿3+6 x2)+(4 x+8)¿ Factor x2 out of the first group of terms. Factor the constants out of both groups. y=x2(3 x+6)+(4 x+8) y=3 x2(x+2)+4( x+2) Add the two terms by adding the coefficients y=3 x2(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 15 he st ex po ne nt g order nt te r m d fo r m 1. f ( x )=4+4 x4+8 x 2. f ( x )=(x+2)(x−2) B. Directions: Write the factored form of the following polynomial functions by completing the table: 1. y=343 x3+27 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+b2)¿ y=¿¿ So, the factored form is y=¿¿) 2. y=27 x3−8 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+b2)¿ y=¿¿ So, the factored form is y=¿¿) 16 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: 1. y=x4−512x 2. y=9x3−36x2+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 t2−0.01t 3. 2. The total number of hexagons in a honeycomb can be modeled by the function f (r )=1+3 r2−3 r. Assessment Directions: Choose the letter that best answers each question. 1. What is the product of (x+3)(x+3)? A. x2+3 x+9 B. x2−3 x+9 C. x2+6 x+9 D. x2−6 x+9 17 3. What is the leading term of f ( x )=x2+4 x3+1? A. x B. 2 C. 3 4 x3 4. How should the polynomial function f ( x )=x4−8 x2+ x 2 + 1 2 +4 x3 be written in standard form? A. f ( x )=−8 x2+ 1 2 +4 x3+ x 4 + x 2 B. f ( x )= x 2 −8 x2+ 1 2 +4 x3+x 4 C. f ( x )=x4+4 x3−8x2+ x 2 +1 2 D. f ( x )=4 x3+1 2 −8x2+ x 2 +x4 5. How should f ( x )=x4+ x3+x2+x be written in factored form? A. f ( x )=x (x+1)(x2+1) B. f ( x )=x (1)(x2+1) C. f ( x )=x (x−1)(x2+1) D. f ( x )=x (−1)¿) 20 21