Class 9 › Mathematics › Ganita Manjari Introduction to Linear Polynomials Class 9 Mathematics · Ganita Manjari · Chapter 2
Practice all → Summary Questions & answers Practise this chapter → This chapter builds on algebraic expressions to define univariate polynomials — expressions in a single variable such as \(5y^3 + y^2 - 8\) — and their degree, the highest power of the variable. Polynomials of degree \(0,1,2,3\) are called constant, linear, quadratic and cubic respectively. The focus is the linear polynomial, of the form \(ax + b\). A linear pattern is a sequence whose consecutive terms differ by a constant amount, and many real situations (taxi fares, savings, growing tile patterns) follow such patterns. When a linear polynomial is set equal to a constant, a linear equation results. Two variables related linearly satisfy \(y = ax + b\): here \(a\) is the slope and \(b\) the \(y\)-intercept. Plotting two points and joining them graphs the line. Changing \(a\) tilts the line (steeper when \(|a|>1\)); changing \(b\) shifts it up or down, keeping lines of equal slope parallel. Positive slope models linear growth, negative slope models linear decay. The chapter also treats polynomials as input-output (function) machines.
Polynomial
An algebraic expression in one variable made of terms with whole-number powers, such as \(x^2 + 5x + 1\).
Degree
The highest power of the variable in a polynomial; e.g. \(3z+7\) has degree \(1\).
Linear polynomial
A polynomial of degree one, of the form \(ax + b\) with \(a \ne 0\).
Slope
The coefficient \(a\) in \(y = ax + b\); it measures the steepness of the line.
\(y\)-intercept
The constant \(b\) in \(y = ax + b\); the point \((0,b)\) where the line meets the \(y\)-axis.
Linear pattern
A sequence in which the difference between consecutive terms is constant.
Q1 What is the degree of the polynomial \(2x^2 - 5x + 3\)? Answer Q2 Find the value of the linear polynomial \(5x - 3\) when \(x = 2\). Answer Q3 What do \(a\) and \(b\) represent in the line \(y = ax + b\)? Answer Q4 The sum of two numbers is \(64\) and one is \(10\) more than the other; find them. Answer Q5 Does linear growth have positive or negative slope? Answer Q6 Through which point does every line \(y = ax\) pass? Answer Q7 Find \(a\) and \(b\) if \(y=ax+b\) gives \(y=350\) at \(x=10\) and \(y=550\) at \(x=20\). Answer Q8 What is the nth term of the tile pattern \(1,3,5,7,\dots\)? Answer Q9 How are two lines with equal slope but different intercepts related? Answer Q10 A phone costing \(\,\)10000 loses \(\,\)800 per year; what models its value after \(t\) years? Answer Q1 1. Can you identify the terms, variables and coefficients of this algebraic expression? Answer Q2 2. How is it different from the algebraic expression in Example 1? Answer Q1 1. Can you identify the terms, variables and coefficients of this algebraic expression? Answer Q2 2. Can you point out any similarity or difference between the algebraic expressions obtained in Examples 1 and 3? Answer Q1 1. Find the degrees of the following polynomials: (i) \(2x^2 - 5x + 3\) Answer Q2 1. Find the degrees of the following polynomials: (ii) \(y^3 + 2y - 1\) Answer Q3 1. Find the degrees of the following polynomials: (iii) \(-9\) Answer Q4 1. Find the degrees of the following polynomials: (iv) \(4z - 3\) Answer Q5 2. Write polynomials of degrees 1, 2 and 3. Answer Q6 3. What are the coefficients of \(x^2\) and \(x^3\) in the polynomial \(x^4 - 3x^3 + 6x^2 - 2x + 7\)? Answer Q7 4. What is the coefficient of \(z\) in the polynomial \(4z^3 + 5z^2 - 11\)? Answer Q8 5. What is the constant term of the polynomial \(9x^3 + 5x^2 - 8x - 10\)? Answer Q1 Find the perimeter of squares with sides 1 cm, 1.5 cm, 2 cm, 2.5 cm and 3 cm. What will happen to the perimeters if the sides increase by 0.5 cm? Answer Q1 If a player paid \(750\), how many matches did he play? Answer Q1 We have learnt that to evaluate the value of an algebraic expression, we substitute a value of the variable in the given expression. Consider Example 3, where the wire is bent to form a rectangle. Here, the area of the rectangle, \(10x - x^2\), is a function of \(x\). Can you interpret this as an input-output process? What value does the expression take when \(x = 6\) cm? Answer Q1 1. Find the value of the linear polynomial \(5x - 3\) if: (i) \(x = 0\) Answer Q2 1. Find the value of the linear polynomial \(5x - 3\) if: (ii) \(x = -1\) Answer Q3 1. Find the value of the linear polynomial \(5x - 3\) if: (iii) \(x = 2\) Answer Q4 2. Find the value of the quadratic polynomial \(7s^2 - 4s + 6\) if: (i) \(s = 0\) Answer Q5 2. Find the value of the quadratic polynomial \(7s^2 - 4s + 6\) if: (ii) \(s = -3\) Answer Q6 2. Find the value of the quadratic polynomial \(7s^2 - 4s + 6\) if: (iii) \(s = 4\) Answer Q7 3. The present age of Salil's mother is three times Salil's present age. After 5 years, their ages will add up to 70 years. Find their present ages. Answer Q8 4. The difference between two positive integers is 63. The ratio of the two integers is 2:5. Find the two integers. Answer Q9 5. Ruby has 3 times as many two-rupee coins as she has five rupee-coins. If she has a total \(88\), how many coins does she have of each type? Answer Q10 6. A farmer cuts a 300 feet fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces? Answer Q11 7. If the length of a rectangle is three more than twice its width and its perimeter is 24 cm, what are the dimensions of the rectangle? Answer Q1 Predict the number of squares in the next three stages of the pattern and write the sequence of numbers up to Stage 7 of the pattern. Answer Q1 Using the expression \(2n - 1\), can you find out how many tiles will be there in the 15th stage and the 26th stage of the pattern? Also, which stage will contain 21 tiles and 47 tiles? Answer Q1 What amount will be left on the 15th day? How many days will it take for the entire amount to be spent? Answer Q1 For how many km will the fare be \(130\)? Answer Q1 1. A student has \(500\) in her savings bank account. She gets \(150\) every month as pocket money. How much money will she have at the end of every month from the second month onwards? Find a linear expression to represent the amount she will have in the \(n^{th}\) month. Answer Q2 2. A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after 1, 2, 3, ... hours? Find a linear expression to represent the number of members at the end of the \(n^{th}\) hour. Answer Q3 3. Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (i) 12 cm. Find the linear pattern representing the area of the rectangle. Answer Q4 3. Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (ii) 10 cm. Find the linear pattern representing the area of the rectangle. Answer Q5 3. Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (iii) 8 cm. Find the linear pattern representing the area of the rectangle. Answer Q6 4. Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (i) 5 cm. Find the linear pattern representing the volume of the rectangular box. Answer Q7 4. Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (ii) 9 cm. Find the linear pattern representing the volume of the rectangular box. Answer Q8 4. Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (iii) 13 cm. Find the linear pattern representing the volume of the rectangular box. Answer Q9 5. Sarita is reading a book of 500 pages. She reads 20 pages every day. How many pages will be left after 15 days? Express this as a linear pattern. Answer Q1 What is the cost for travelling 15 km? For how many kilometres will the cost of the journey be \(700\)? Answer Q1 What will be the height of the water at the end of 5 months? Answer Q1 1. Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month. (i) Find the height after 7 months. Answer Q2 1. Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month. (ii) Make a table of values for \(t\) varying from 0 to 10 months and show how the height, \(h\), increases every month. Answer Q3 1. Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month. (iii) Find an expression that relates \(h\) and \(t\), and explain why it represents linear growth. Answer Q4 2. A mobile phone is bought for \(10,000\). Its value decreases by \(800\) every year. (i) Find the value of the phone after 3 years. Answer Q5 2. A mobile phone is bought for \(10,000\). Its value decreases by \(800\) every year. (ii) Make a table of values for \(t\) varying from 0 to 8 years and show how the value of the phone, \(v\), depreciates with time. Answer Q6 2. A mobile phone is bought for \(10,000\). Its value decreases by \(800\) every year. (iii) Find an expression that relates \(v\) and \(t\), and explain why it represents linear decay. Answer Q7 3. The initial population of a village is 750. Every year, 50 people move from a nearby city to the village. (i) Find the population of the village after 6 years. Answer Q8 3. The initial population of a village is 750. Every year, 50 people move from a nearby city to the village. (ii) Make a table of values for \(t\) varying from 0 to 10 years and show how the population, \(P\), increases every year. Answer Q9 3. The initial population of a village is 750. Every year, 50 people move from a nearby city to the village. (iii) Find an expression that relates \(P\) and \(t\), and explain why it represents linear growth. Answer Q10 4. A telecom company charges \(600\) for a certain recharge scheme. This prepaid balance is reduced by \(15\) each day after the recharge. (i) Write an equation that models the remaining balance \(b(x)\) after using the scheme for \(x\) days. Explain why it represents linear decay. Answer Q11 4. A telecom company charges \(600\) for a certain recharge scheme. This prepaid balance is reduced by \(15\) each day after the recharge. (ii) After how many days will the balance run out? Answer Q12 4. A telecom company charges \(600\) for a certain recharge scheme. This prepaid balance is reduced by \(15\) each day after the recharge. (iii) Make a table of values for \(x\) varying from 1 to 10 days and show how the balance \(b(x)\), reduces with time. Answer Q1 Can you guess what the numbers 20 and 150 in the equation \(y = 20x + 150\) represent? Answer Q1 1. A learning platform charges a fixed monthly fee and an additional cost per digital learning module accessed. A student observes that when she accessed 10 modules, her bill was \(400\). When she accessed 14 modules, her bill was \(500\). If the monthly bill \(y\) depends on the number of modules accessed, \(x\), according to the relation \(y = ax + b\), find the values of \(a\) and \(b\). Answer Q2 2. A gym charges a fixed monthly fee and an additional cost per hour for using the badminton court. A student using the gym observed that when she used the badminton court for 10 hours, her bill was \(800\). When she used it for 15 hours, her bill was \(1100\). If the monthly bill \(y\) depends on the hours of the use of the badminton court, \(x\), according to the relation \(y = ax + b\), find the values of \(a\) and \(b\). Answer Q3 3. Consider the relationship between temperature measured in degrees Celsius (\(°C\)) and degrees Fahrenheit (\(°F\)), which is given by \(°C = a °F + b\). Find \(a\) and \(b\), given that ice melts at 0 degrees Celsius and 32 degrees Fahrenheit, and water boils at 100 degrees Celsius and 212 degrees Fahrenheit. Answer Q1 Identify other points on the line by completing the following table. \(x\) 1 2 5 7 9 12 20 \(y\) 3 15 Answer Q1 Differentiate between the graphs of the equations \(y = 3x + 1\), and \(y = -3x + 1\). Answer Q1 Does this help you to conclude anything about the linear equation \(y = ax + b\) when \(a\) is fixed but \(b\) varies? Answer Q1 1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (i) \(y = 4x\), \(y = 2x\), \(y = x\) Answer Q2 1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (ii) \(y = -6x\), \(y = -3x\), \(y = -x\) Answer Q3 1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (iii) \(y = 5x\), \(y = -5x\) Answer Q4 1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (iv) \(y = 3x - 1\), \(y = 3x\), \(y = 3x + 1\) Answer Q5 1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (v) \(y = -2x - 3\), \(y = -2x\), \(y = 2x + 3\) Answer Q1 1. Write a polynomial of degree 3 in the variable \(x\), in which the coefficient of the \(x^2\) term is \(-7\). Answer Q2 2. Find the values of the following polynomials at the indicated values of the variables. (i) \(5x^2 - 3x + 7\) if \(x = 1\) Answer Q3 2. Find the values of the following polynomials at the indicated values of the variables. (ii) \(4t^3 - t^2 + 6\) if \(t = a\) Answer Q4 3. If we multiply a number by \(\frac{5}{2}\) and add \(\frac{2}{3}\) to the product, we get \(\frac{-7}{12}\). Find the number. Answer Q5 4. A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers? Answer Q6 5. If you have \(800\) and you save \(250\) every month, find the amount you have after (i) 6 months. Express this as a linear pattern. Answer Q7 5. If you have \(800\) and you save \(250\) every month, find the amount you have after (ii) 2 years. Express this as a linear pattern. Answer Q8 6. The digits of a two-digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. Find both the numbers. Answer Q9 7. Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis. (i) \(y = -3x + 4\) Answer Q10 7. Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis. (ii) \(2y = 4x + 7\) Answer Q11 7. Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis. (iii) \(5y = 6x - 10\) Answer Q12 7. Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis. (iv) \(3y = 6x - 11\) Answer Q13 8. If the temperature of a liquid can be measured in Kelvin units as \(x\) K and in Fahrenheit units as \(y\) °F, the relation between the two systems of measurement of temperature is given by the linear equation \(y = \frac{9}{5}(x - 273) + 32\). (i) Find the temperature of the liquid in Fahrenheit if the temperature of the liquid is 313 K. Answer Q14 8. If the temperature of a liquid can be measured in Kelvin units as \(x\) K and in Fahrenheit units as \(y\) °F, the relation between the two systems of measurement of temperature is given by the linear equation \(y = \frac{9}{5}(x - 273) + 32\). (ii) If the temperature is 158 °F, then find the temperature in Kelvin. Answer Q15 9. The work done by a body on the application of a constant force is the product of the constant force and the distance travelled by the body in the direction of the force. Express this in the form of a linear equation in two variables (work \(w\) and distance \(d\)), and draw its graph by taking the constant force as 3 units. What is the work done when the distance travelled is 2 units? Verify it by plotting it on the graph. Answer Q16 10. The graph of a linear polynomial \(p(x)\) passes through the points (1, 5) and (3, 11). (i) Find the polynomial \(p(x)\). Answer Q17 10. The graph of a linear polynomial \(p(x)\) passes through the points (1, 5) and (3, 11). (ii) Find the coordinates where the graph of \(p(x)\) cuts the axes. Answer Q18 10. The graph of a linear polynomial \(p(x)\) passes through the points (1, 5) and (3, 11). (iii) Draw the graph of \(p(x)\) and verify your answers. Answer Q19 11. Let \(p(x) = ax + b\) and \(q(x) = cx + d\) be two linear polynomials such that: (i) \(p(0) = 5\). (ii) The polynomial \(p(x) - q(x)\) cuts the x-axis at (3, 0). (iii) The sum \(p(x) + q(x)\) is equal to \(6x + 4\) for all real \(x\). Find the polynomials \(p(x)\) and \(q(x)\). Answer Q20 12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. Stage 1 Stage 2 Stage 3 (i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages? Answer Q21 12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. (ii) Complete the following table. Stage Number 1 2 3 4 5 ... n Number of matchsticks Answer Q22 12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. (iii) Find a rule to determine the number of matchsticks required for the nth stage. Answer Q23 12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. (iv) How many matchsticks will be required for the 15th stage of the pattern? Answer Q24 12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. (v) Can 200 matchsticks form a stage in this pattern? Justify your answer. Answer Q25 13. Let \(p(x) = ax + b\) and \(q(x) = cx + d\) be two linear polynomials such that: (i) The graph of \(p(x)\) passes through the points (2, 3) and (6, 11). (ii) The graph of \(q(x)\) passes through the point (4, -1). (iii) The graph of \(q(x)\) is parallel to the graph of \(p(x)\). Find the polynomials \(p(x)\) and \(q(x)\). Also, find the coordinates of the point where these lines meet the x-axis. Answer Q26 14. What do all linear functions of the form \(f(x) = ax + a\), \(a > 0\), have in common? Answer
What is the degree of the polynomial \(2x^2 - 5x + 3\)?
A \(3\) B \(2\) C \(0\) D \(1\)
Flashcards & interactive practice 
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