STANDARD IX

MATHEMATICS

THE NATIONAL ANTHEM Jana-gana-mana adhinayaka, jay hey Bharatha-bhagya-vidhata. Punjab-Sindh-Gujarat-Maratha Dravida-Utkala-Banga Vindhya-Himachala-Yamuna-Ganga Uchchala-Jaladhi-taranga Tava subha name jage, Tava subha asisa mage, Gahe tava jaya gatha. Jana-gana-mangala-dayaka jaya he Bharatha-bhagya-vidhata. Jaya he, jaya he,jaya he, Jaya jaya jaya, jaya he!

PLEDGE India is my country. All Indians are my brothers and sisters. I love my country, and I am proud of its rich and varied heritage. I shall always strive to be worthy of it. I shall give respect to my parents, teachers and all elders and treat everyone with courtesy. I pledge my devotion to my country and my people. In their well-being and prosperity alone lies my happiness.

Dear children, The elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. Rene Descartes, popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables. In this book we discuss about polynomials. Degree of the polynomial, general form of polynomial and problems related to it are discussed in this book. Polynomials are an important part of the "language" of mathematics and algebra. They are uesd in nearly every field of mathematics to express numbers as a result of mathematical operations. Polynomials are also "building blocks" in other types of mathematical expressions, such as rational expressions. This book helps you to get acquaint with polynomials.

With Love AUTHOR

Interesting facts about Mathematics 1. Every odd numbers has an 'e' in it. Odd numbers need to end in 1,3,5,7,9 and each of the numbers have an 'e' in it. 2. There is no zero(0) in Roman numerals. The entire number system is generated by the seven different letters, namely I, V,X, L, C, D and M. Using these we can write numbers up to 3999. 3. In the entire Hindu Arabic number system, there is only one number which can be spelled with the same number of letters as itself. That number is FOUR. 4. In the decimal number system, there are 10 digits from 0 to 9. It is also known as the Hindu Arabic numeral system, invented more than 1000 years ago. 5. Forty is the only number that is spelt with letters arranged in alphabetical order. 6. One is the only number that is spelt with letters arranged in descending order. 7. The opposite sides of a die always add up to 7. For example, 6 and 1 will always be on opposite sides which add up to7. 8. 2 is the only natural number satisfying 2+2 = 2x2

CONTENT 1.1 Introduction

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1.2 What is a polynomial?

4

1.3 Degree of a polynomial.

4

1.4 Terms and coefficient of a polynomial

5

1.5 What cannot be a polynomial?

6

1.6 Polynomial Operations

6

1.7 Types of polynomial

9

1.8 Finding the value of a polynomial

12

1.9 Solving problems.

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Right to Education Act Article 21-A and the RTE Act came into effect on 1 April 2010. The title of the RTE Act incorporates the words ‘free and compulsory’. ‘Free education’ means that no child, other than a child who has been admitted by his or her parents to a school which is not supported by the appropriate Government, shall be liable to pay any kind of fee or charges or expenses which may prevent him or her from pursuing and completing elementary education. ‘Compulsory education’ casts an obligation on the appropriate Government and local authorities to provide and ensure admission, attendance and completion of elementary education by all children in the 6-14 age group. With this, India has moved forward to a rights based framework that casts a legal obligation on the Central and State Governments to implement this fundamental child right as enshrined in the Article 21A of the Constitution, in accordance with the provisions of the RTE Act.

1.1 INTRODUCTION Sneha go for a shop for buying pen. Then she saw an offer board.

BUY 1 PACKET PEN GET 2 PEN FREE

1 packet contains 4 pens. Then how many pens she got when she buy 2 packets? 1packet contains 4 pens. With 1 packet 2 pens are get free. So number of pens she got = 4+2 = 6 This computation is quicker isn‟t it? If she buys 2 packet of pens. Then how many pens she got?

She got 6 pens when she buy 1 packet of pen. so number of pens she got when buying 2 packets = 6×2 = 12. Isn‟t it?

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Then how many pens she got when she buy 10 packets of pen? 6×10 = 60. Is it correct? Then how many pens she got when she buy x packets of pen? 6 × 𝑥 = 6𝑥. It is very easy to compute, isn‟t it? You are familiar with this type of questions in the previous standard. Now let us observe two situations:  A plot is in the shape of a triangle. The longest side is 3 times the middle side and smallest side is 2 units shorter than the middle side. Let y represent the length of the middle side, then what is the perimeter in terms of y?  The length of a rectangular hall is twice its breadth. Let x represent the breadth of the hall. What is the area of the floor of the hall in terms of x? In the above situations, there is an unknown in each statement. In the first situation, middle side is given as „y‟ units.

Since, Perimeter of triangle = sum of all sides Perimeter = y + 3y + y – 2 = 5y– 2

If the perimeter of triangle is denoted as P, then how can you write it?

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P = 5y-2 . Isn‟t it? Here the variable is y. So this can be generally written as P(y) = 5y-2. If the length of the middle side is „z‟ units. What is the perimeter in terms of z? Perimeter = z + 3z + z - 2 = 5z-2 How can you write it in the general form? P(z) = 5z-2

Isn‟t it?

Here the variable is z. so it is written as P(z). Similarly in the second situation, length is given as twice the breadth.

So, if breadth = x, length = 2x Since area of rectangle = lb

Area = (2x)×(x) = 2𝑥 2 If the area of rectangle is denoted as a, then how can you write it? It is a(x) = 2𝑥 2 You have learned about algebraic expressions in the previous class, haven't you ?

In this chapter we are going to discuss about Polynomials. Then, what is a Polynomial? Let‟s take a look.

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1.2 What is a Polynomial? Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations such as addition,subtraction, multiplication and also positive integer exponents for polynomial expressions but not division by variable.

Did you know? The word Polynomial comes from Greek words ‘poly' means 'many' and 'nominal' means 'terms'. So altogether it is said as ''many terms”.

Eg: 1) 𝑥 2 + 𝑥 − 12 2) x – 6

Notation The polynomial function is denoted by P(x) where x represents the variable. For example, P(x) = 𝑥 2 − 5𝑥 + 11 If the variable is denoted by a, then the function will be P(a).

1.3 DEGREE OF A POLYNOMIAL The degree of a polynomial is defined as the highest exponent of a monomial within a polynomial. Thus, a polynomial equation having one variable which has the largest exponent is called a degree of the polynomial. So, the degree of the polynomial 3𝑥 7 − 4𝑥 6 + 𝑥 + 9 is 7 and the degree of the polynomial 5𝑦 6 − 4𝑦 2 − 6 is 6. The degree of a non-zero constant polynomial is zero. Consider the polynomial 3x+5. It is of degree 1. So it is called first degree polynomial(Linear polynomial).

Next, consider the polynomial 𝑥 2 + 5𝑥 + 4. It is of degree 2. So it is called second degree polynomial(Quadratic Polynomial).

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Now, consider the polynomial 5𝑥 3 − 4𝑥 2 + 𝑥 − 1. It is of degree 3. So it is called Third degree polynomial(Cubic Polynomial). Based on the degree, we can write the general forms of all polynomials. First degree polynomial

: ax + b

Second degree polynomial : 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Third degree polynomial

: 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑

Here the letters a,b,c and d denote fixed numbers. A type of polynomial where all variable‟s coefficients are equal to zero are called Zero polynomial.

Example 1: Find the degree of the polynomial P(x) = 6𝑥 4 + 3𝑥 2 + 5𝑥 + 19 ? Solution: As the highest power of the variable is 4, the degree of the polynomial is 4.

1.4 Terms AND COEFFICIENT of a Polynomial The terms of polynomials are the parts of the expression that are generally separated by “+” or “-” signs. So, each part of a polynomial in an expression is a term. For example, in a polynomial, say, 2𝑥 2 + 5𝑥 + 4, the number of terms will be 3. The classification of a polynomial is done based on the number of terms in it. Each term of a polynomial has a coefficient. So, in −𝑥 3 + 4𝑥 2 + 7𝑥 − 2, the coefficient of 𝑥 3 is -1, the coefficient of 𝑥 2 is 4, the coefficient of x is 7 and -2 is the coefficient of 𝑥 0 . (Remember , 𝑥 0 =1)

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1.5 WHAT CANNOT BE A POLYNOMIAL?  If any term has a variable with a negative exponent, then the expression is not a polynomial. Eg: x+4+𝑥 −3  A term with division by a variable is equivalent to a negative exponent, so such an expression is not a polynomial. Eg:

5 𝑥2

= 5𝑥 −2

 If a term has a variable with a fractional exponent (that does not reduce to a whole number), the expression is not a polynomial. Eg: 2+7x+𝑥 1/3  A term with a variable in a radical is equivalent to a fractional exponent, so such an expression is not a polynomial. Eg: 4-3x+ 𝑥

DO THIS State which of the following are polynomials and which are not? Give reasons. (i) 2𝒙𝟑

(ii)

𝟏

(iii) 4𝒛𝟐

𝒙−𝟏

+

𝟏 𝟕

1.6 POLYNOMIAL OPERATIONS There are four main polynomial operations which are:    

Addition of Polynomials Subtraction of Polynomials Multiplication of Polynomials Division of Polynomials

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(iv) 𝒑−𝟐 + 𝟏

Addition of Polynomials To add polynomials, always add the like terms. That is, the terms having the same variable and power. The addition of polynomials always results in a polynomial of the same degree. Example 2: Find the sum of two polynomials: 5𝑥 3 + 3𝑥 2 𝑦 + 4𝑥𝑦 − 6𝑦 2 and 3𝑥 2 + 7𝑥 2 𝑦 − 2𝑥𝑦 + 4𝑥𝑦 2 − 5 ? Solution: First combine the like terms while leaving the unlike terms as they are. Hence

3𝑥 3 + 3𝑥 2 𝑦 + 4𝑥𝑦 − 6𝑦 2 + 3𝑥 2 + 7𝑥 2 𝑦 − 2𝑥𝑦 + 5𝑥𝑦 2 − 5 = 3𝑥 3 + 3𝑥 2 + 𝑥 2 𝑦 3 + 7 + 𝑥𝑦 4 − 2 + 5𝑥𝑦 2 − 6𝑦 2 − 5 = 3𝑥 3 + 3𝑥 2 + 10𝑥 2 𝑦 + 2𝑥𝑦 + 5𝑥𝑦 2 − 6𝑦 2 − 5 Subtraction of Polynomials Subtracting polynomials is similar to addition, the only difference being the type of operation. So, subtract the like terms to obtain the solution. It should be noted that subtraction of polynomials also results in a polynomial of the same degree. Example 3: Find the difference of two polynomials: 9x3+3x2y+4xy−6y2 , 3x2+7x2y−2xy+8xy2−5 Solution:

First, combine the like terms while leaving the unlike terms as they are. Hence, (9x3+3x2y+4xy−6y2)−(3x2+7x2y−2xy+8xy2−5 7

= 9x3−3x2+x2y 3−7 + xy(4+2) −8xy2−6y2+5 = 9x3−3x2−4x2y+6xy−8xy2−6y2+5 Multiplication of Polynomials

Two or more polynomial when multiplied always result in a polynomial of higher degree (unless one of them is a constant polynomial). Example 4: Solve 6x−3y × 2x+5y Solution: ⇒ 6x ×(2x+5y)–3y × (2x+5y) ( Using distributive law of multiplication) ⇒ (12x2+30xy) – (6yx+15y2) ( Using distributive law of multiplication) ⇒12x2+30xy–6xy–15y2 (as xy = yx) Thus, 6x−3y × 2x+5y =12x2+24xy−15y2 Division of Polynomials Division of two polynomial may or may not result in a polynomial. To divide polynomials, follow the given steps:

Polynomial Division Steps: If a polynomial has more than one term, we use long division method for the same. Following are the steps for it. 1. 2. 3. 4. 5. 6.

Write the polynomial in descending order. Check the highest power and divide the terms by the same. Use the answer in step 2 as the division symbol. Now subtract it and bring down the next term. Repeat steps 2 to 4 until you have no more terms to carry down. Note the final answer, including remainder, will be in the fraction form (last subtract term).

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Example 5: Divide the polynomial 6x3 + 150x2 + 5x by 15x. Solution: From the given, Dividend = 6x3 + 150x2 + 5x Divisor = 15x Here, the degree of dividend, i.e. 3, is greater than the degree of the divisor, i.e. 1. So,

6𝑥 3 +150𝑥 2 +5𝑥 15𝑥

Now divide each term of the numerator by 15 x. 6𝑥 3 15𝑥

+

150𝑥 2 15𝑥

+

5𝑥 15𝑥

2

1

3

3

= x2 + 10x +

1.7 Types of Polynomial

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Exercise 1.1

1) What are the coefficients of y in the following expressions ? 4x-8y , 8 + yz , ny + n , 𝑦𝑧 2 + 6 2) Identify the terms in the following polynomials (a) x-5

(b) 8+x+𝑥 2

(c) y - 𝑦 3

(d) 5𝑥𝑦 2 + 9𝑥 2 𝑦

(e) -ab+2𝑏 2 − 3𝑎2

(f) 4x+5

(g) -7x+5y

(h) xy+2𝑥 2 𝑦 2

(i)pq+q

3) Identify the numerical coefficients of terms (other than constants) in the following expressions: (i) x+2xy+5y

(ii) 8-3𝑡 2

(iv) 1 + 𝑡 + 𝑡 2 + 𝑡 3

(v) −𝑝2 𝑞 2 + 8𝑝𝑞

(iii) 100n+1000k

4) Identify terms which contain 𝑦 2 and give the coefficient of 𝑦 2 ? a) 8 − 𝑥𝑦 2

b) 4𝑥 2 𝑦 − 12𝑥𝑦 2 + 7𝑦 2

c) 12𝑦 2 + 8𝑥

d) 22𝑥 2 𝑦 2 + 15𝑥𝑦

5) Simplify: a) 18b-32+14b-21b b) 𝑞 − 𝑞 − 𝑝 − 𝑝 − 𝑝 − 𝑞 c) 3𝑏 − 2𝑎 − 𝑎𝑏 − 𝑏 − 𝑎 + 𝑎𝑏 + 3𝑎𝑏 + 𝑎 − 𝑏

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8) Multiply: (i) 5 − 2𝑥 3 + 𝑥

(ii) 𝑥 + 7𝑦 7𝑥 − 𝑦

(iii) 𝑎2 + 𝑏 𝑎 + 𝑏 2

(iv) 𝑝2 − 𝑞 2 2𝑝 + 𝑞

(v) 𝑥 2 − 5 𝑥 + 5

(v) 𝑡 + 𝑠 2 𝑡 2 − 𝑠

9) Divide: (i) 5 2𝑥 + 1 3𝑥 + 5 ÷ 2𝑥 + 1 (ii) 10𝑥 − 25 ÷ 2𝑥 − 5 (iii) 66𝑝2 𝑞 2 𝑟 ÷ 11𝑝𝑞

10) Classify into monomials, binomials and trinomials. a) 4𝑦 − 8𝑧

b) 𝑥 + 𝑦 + 𝑥𝑦

c) 𝑎𝑏 − 𝑎 + 𝑏

c) 𝑧 2 − 3𝑧 + 9

d) 𝑦 2

e) 1 + 𝑥 − 𝑥 2

11) What should be taken away from 3𝑥 2 − 4𝑦 2 + 5𝑥𝑦 + 20 to obtain −𝑥 2 + 𝑦 2 + 6𝑥𝑦 + 20 ? 12) (a) What should be added to 𝑥 2 + 𝑥𝑦 + 𝑦 2 to obtain 2𝑥 2 + 3𝑥𝑦 ? (b) What should be subtracted from 2𝑏 + 8𝑎 + 9 to get −3𝑏 + 7𝑎 + 14 ?

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1.8 Finding the value of a Polynomial There are number of situations in which we need to find the value of an expression. Such aswhen we wish to check whether a particular value of a variable satisfies a given polynomial or not. For example, the area of a square is 𝑙 2 , where l is the length of a side of the square. If l=6 cm, the area is 62 𝑐𝑚2 or 36 𝑐𝑚2 . If the side is 12 cm, the area is 122 𝑐𝑚2 or 144 𝑐𝑚2 and so on. Now consider polynomials of two variables. For example, x + y. The value of 𝑥 + 𝑦 for 𝑥 = 4 and 𝑦 = 8 is 4 + 8 = 12. Example 6: Find the value of the following polynomials for 𝑎 = 5, 𝑏 = 4 (i 8𝑎 − 5𝑏

(iii)𝑎3 + 8𝑎𝑏 + 𝑏 2

(ii) a-b

Solution: Substituting a = 5 and b = 4 in (i) 8a – 5b, we get

8𝑎 – 5𝑏 = 8 × 5 – 5 × 4 = 40 − 20 = 20 (ii) 𝑎 − 𝑏, we get 𝑎−𝑏 =5−4=1 (iii) 𝑎3 + 8𝑎𝑏 + 𝑏 2 , we get 𝑎3 + 8𝑎𝑏 + 𝑏 2 = 53 + 8 × 5 × 4 + 42

= 125+160+16 = 301

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EXERCISE 1.2

1) If m = -2, find the value of: (i) 5m+7

(ii) –m+2

(iv) −6𝑚2 + 8𝑚 + 5

(v) −7𝑚3 − 4𝑚2 + 10𝑝 + 2

(iii) 3m-9

2) Simplify the expressions and find their values if x= 4, a = -3, b = -1. (i 7𝑥 − 8 − 𝑥 + 6

(ii) 5 − 8𝑥 + 4𝑥 − 3

(iii) 2𝑎 + 9 − 8𝑎 + 1

(iv) 12 − 4𝑏 − 7 − 9𝑏

(v) 6𝑎 − 6𝑏 − 9 − 8 + 2𝑎

(vi) 6𝑥 + 5 𝑎 − 𝑏

3) What should be the value of a if the value of 2𝑥 2 + 𝑥 − 𝑎 equals to 5, when x = 0 ? 4) Simplify the expression 2 𝑎2 + 𝑎𝑏 + 3 − 𝑎𝑏 and find its value when a = 8 and b = 3 ? 5) (i) If z = 25, find the value of 𝑧 3 − 4 𝑧 − 20 (ii) If p = -15, find the value of 𝑝2 − 2𝑝 − 80

1.9 Solving Problems We can solve a problem by representing it as a polynomial and then find the value of the variable(s) in the polynomial equation.

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Example 7: The perimeter of a rectangle is 104 m and its area is 640 𝑚2 . Find its length and breadth ? Solution: Let the length and breadth of the rectangle be x metre and y metre. Perimeter, p(x) = 2(x + y) ∴ 104 = 2 𝑥 + 𝑦 𝑥 + 𝑦 = 52 𝑦 = 52 – 𝑥 Area = 640 𝑚2 ∴ 𝑥𝑦 = 640 𝑥 52 − 𝑥 = 640 𝑥 2 − 52𝑥 + 640 = 0 𝑥 2 − 32𝑥 − 20𝑥 + 640 = 0 𝑥 𝑥 − 32 − 20 𝑥 − 32 = 0 𝑥 − 32 𝑥 − 20 = 0 𝑥 = 32, 20 When x = 32, y = 52 – 32 =20 When x = 20, y = 52 – 20 =32 Thus, the length and breadth of the rectangle are 32 m and 20 m

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Example 8: The hypotenuse of a right-angled triangle is 26 cm and the sum of other two sides is 34 cm. Find the lengths of its sides ? Solution: Hypotenuse = 26 cm The sum of other two sides is 34 cm. So, let the other two sides be x cm and (34 – x) cm. Using pythagoas theorem, 26

2

= 𝑥 2 + 34 − 𝑥

2

676 = 𝑥 2 + 𝑥 2 + 1156 − 68𝑥 2𝑥 2 − 68𝑥 + 480 = 0 𝑥 2 − 34𝑥 + 240 = 0 𝑥 2 − 10𝑥 − 24𝑥 + 240 = 0 𝑥 𝑥 − 10 − 24 𝑥 − 10 = 0 𝑥 − 10 𝑥 − 24 = 0 𝑥 = 10, 24 When 𝑥 = 10, 34 – 𝑥 = 24 When 𝑥 = 24, 34 − 𝑥 = 10 Thus, the lengths of the three sides of the right-angled triangle are 10 cm, 24 cm and 26 cm.

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Example 9: The ages of two sisters are 11 years and 14 years. In how many years time will the product of their ages be 304 ? Solution: The ages of two sisters are 11 years and 14 years. Let in x number of years the product of their ages be 304. ∴ 11 + 𝑥 14 + 𝑥 = 304 154 + 11𝑥 + 14𝑥 + 𝑥 2 = 304 𝑥 2 + 25𝑥 − 150 = 0 𝑥 + 30 𝑥 − 5 = 0 𝑥 = −30, 5 But the number of years cannot be negative. So, x = 5. Hence, the required number of years is 5 years.

Try This 1)Find two consecutive positive odd numbers, the sum of whose squares is 74?

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EXERCISE 1.3 1) The hypotenuse of a right -angled triangle exceeds one side by 1cm

and the other side by 18 cm; find the lengths of the sides of the triangle ? 2) One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son‟s age. Find their present ages ? 3) The sum of the squares of two positive integers is 208.If the square of the large number is 18 times the smaller. Find the numbers ? 4) A foot path of uniform width runs round the inside of a rectangular field 32 m long and 24 m wide. If the path occupies 208 𝑚2 , find the width of the footpath ? 5) The area of a big rectangular room is 300 𝑚2 . If the length were decreased by 5 m and the breadth increased by 5m; the area would

be unaltered. Find the length of the room ?

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SUPPLEMENTARY KNOWLEDGEt

HISTORY OF POLYNOMIALS

NOTES

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