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Introduction to Linear Polynomials

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Chapter Notes

Overview
This chapter introduces the concept of polynomials, focusing specifically on linear polynomials in one variable. It builds on earlier knowledge of algebraic expressions, defining terms, variables, coefficients, and degrees. The big idea is that linear polynomials (degree 1) model constant-rate growth or decay, can be expressed as equations y = ax + b, and are visualized as straight lines on a graph, where 'a' is the slope and 'b' is the y-intercept.
Key Concepts (19)
Algebraic expression
A combination of numbers, variables, and operation symbols.
e.g. 4x + 5y + 3 is an algebraic expression where 4x, 5y, and 3 are terms.
Variable
A letter-number used to represent an unknown value; also called a variable.
e.g. In 4x + 5y + 3, x and y are variables.
Term
A part of an algebraic expression separated by + or - signs.
e.g. In 4x + 5y + 3, the terms are 4x, 5y, and 3.
Coefficient
The numerical factor of a term containing a variable.
e.g. In 4x, the coefficient of x is 4; in 5y, the coefficient of y is 5.
Constant term
A term in an algebraic expression that does not contain any variable.
e.g. In 4x + 5y + 3, the constant term is 3.
Polynomial (univariate polynomial)
An algebraic expression involving one variable and its non-negative integer powers.
e.g. x² + 5x + 1 is a polynomial in x; 5y³ + y² + 2y – 1 is a polynomial in y.
Degree of a polynomial
The highest power of the variable in a polynomial.
e.g. In 5y³ + y² + 2y – 1, the degree is 3.
Linear polynomial
A polynomial of degree 1.
e.g. 3z + 7 is a linear polynomial; 4x is a linear polynomial.
Quadratic polynomial
A polynomial of degree 2.
e.g. x² + 5x + 1 is a quadratic polynomial.
Cubic polynomial
A polynomial of degree 3.
e.g. 5y³ + y² + 2y – 1 is a cubic polynomial.
Constant polynomial
A polynomial of degree 0, which can be written as a constant times x⁰.
e.g. 8 is a constant polynomial because it can be written as 8x⁰.
Linear equation
An equation formed by equating a linear polynomial to a constant.
e.g. 2x + 10 = 64 is a linear equation obtained from the linear polynomial 2x + 10.
Linear pattern
A sequence of numbers where the difference between consecutive terms is constant.
e.g. 1, 3, 5, 7, 9, … is a linear pattern with constant difference 2.
Linear growth
A linear pattern where a quantity increases by a constant amount over equal intervals.
e.g. C(d) = 100 + 60d: cost increases by ₹60 for every additional km.
Linear decay
A linear pattern where a quantity decreases by a constant amount over equal intervals.
e.g. h(t) = 3 – 0.5t: water height decreases by 0.5 m every month.
Linear relationship
A relationship between two variables x and y that can be expressed as y = ax + b.
e.g. y = 2x – 1 represents the linear relationship between term number and number of square tiles.
Slope
The constant 'a' in the equation y = ax + b; it represents the steepness of the line and the constant rate of change.
e.g. In y = 2x + 1, the slope is 2.
y-intercept
The constant 'b' in the equation y = ax + b; it is the y-coordinate of the point where the line cuts the y-axis (0, b).
e.g. In y = 2x + 5, the y-intercept is 5, so the line cuts the y-axis at (0, 5).
Parallel lines
Lines that have the same slope but different y-intercepts; they never intersect.
e.g. y = 2x – 1, y = 2x + 1, and y = 2x + 5 are parallel lines.
Formulas & Equations (5)
y = ax + b
Reads as: y equals a x plus b
Variables: y = dependent variable (output), x = independent variable (input), a = slope (rate of change), b = y-intercept (value of y when x = 0)
Use when: To represent any linear relationship between two variables, to find the equation of a line given slope and intercept, or to model linear growth/decay.
⚠ Students often confuse which term is the slope and which is the intercept, or forget that 'b' is the value of y when x = 0, not when x = 1.
y = ax
Reads as: y equals a x
Variables: y = dependent variable, x = independent variable, a = slope
Use when: To represent a linear relationship that passes through the origin (0, 0), e.g., perimeter of a square = 4x.
⚠ Students forget that when b = 0, the line passes through the origin, and they incorrectly add a constant term.
°C = a °F + b
Reads as: degrees Celsius equals a times degrees Fahrenheit plus b
Variables: °C = temperature in Celsius, °F = temperature in Fahrenheit, a and b are constants to be determined
Use when: To convert between Celsius and Fahrenheit scales using given fixed points (0°C = 32°F, 100°C = 212°F).
⚠ Students incorrectly substitute the values or solve the simultaneous equations wrongly, often mixing up which variable corresponds to which scale.
y = (9/5)(x – 273) + 32
Reads as: y equals nine-fifths times (x minus 273) plus 32
Variables: y = temperature in Fahrenheit (°F), x = temperature in Kelvin (K)
Use when: To convert temperature from Kelvin to Fahrenheit.
⚠ Students forget to subtract 273 before multiplying by 9/5, or incorrectly apply the order of operations.
Work = Force × Distance
Reads as: Work equals force times distance
Variables: Work = work done (units depend on context), Force = constant force applied (units), Distance = distance travelled in the direction of force (units)
Use when: To express work done by a constant force as a linear equation in two variables (work w and distance d).
⚠ Students forget that the force must be constant and the distance must be in the direction of the force; they may also omit units.
How It Works
Identifying terms, variables, coefficients, and constant in an algebraic expression
  1. Step 1: Look at the algebraic expression and identify the parts separated by + or - signs — each part is a term.
  2. Step 2: Identify the letter-numbers (variables) in each term.
  3. Step 3: The numerical factor multiplying the variable in a term is the coefficient of that variable.
  4. Step 4: Any term without a variable is the constant term.
💡 Students often forget that the sign before a term belongs to that term; e.g., in 5y³ + y² + 2y – 1, the constant term is –1, not 1.
Finding the degree of a polynomial
  1. Step 1: Identify the variable in the polynomial.
  2. Step 2: Look at the exponent (power) of the variable in each term.
  3. Step 3: The highest exponent among all terms is the degree of the polynomial.
💡 For a constant polynomial like –9, the degree is 0 because it can be written as –9x⁰. Students often mistakenly say it has no degree or degree 1.
Evaluating a polynomial for a given value of the variable
  1. Step 1: Replace the variable in the polynomial with the given numerical value.
  2. Step 2: Perform the arithmetic operations (multiplication, addition, subtraction) in the correct order.
  3. Step 3: The result is the value of the polynomial at that variable value.
💡 Students often make sign errors when substituting negative numbers, e.g., for x = –6 in 2x + 3, they write 2 × –6 + 3 = –12 + 3 = –9, but some incorrectly write 2 × 6 + 3 = 15.
Finding a linear relationship from two data points (y = ax + b)
  1. Step 1: Write two equations by substituting the given (x, y) pairs into y = ax + b.
  2. Step 2: Solve the two equations simultaneously. For example, express b from one equation and substitute into the other to find a.
  3. Step 3: Substitute the value of a back into either equation to find b.
  4. Step 4: Write the final linear equation as y = ax + b.
💡 Students often make algebraic errors when solving simultaneous equations, especially when subtracting terms incorrectly.
Plotting a linear equation on a graph
  1. Step 1: Identify at least two points that satisfy the equation y = ax + b. For example, choose x = 0 to get y = b (point (0, b)), and choose another convenient x value to get a second point.
  2. Step 2: Plot these points on the coordinate plane (x-axis and y-axis).
  3. Step 3: Use a ruler to draw a straight line through the points, extending it in both directions.
  4. Step 4: Label the line with its equation.
💡 Students often plot only one point or plot points incorrectly. They must ensure the line is straight and extends beyond the plotted points.
Identifying slope and y-intercept from a linear equation
  1. Step 1: Write the equation in the form y = ax + b (solve for y if necessary).
  2. Step 2: The coefficient of x (a) is the slope.
  3. Step 3: The constant term (b) is the y-intercept; the line cuts the y-axis at (0, b).
💡 If the equation is not in y = ax + b form, e.g., 2y = 4x + 7, students must first divide by 2 to get y = 2x + 3.5. They often forget to divide the constant term as well.
Recognizing linear growth and linear decay from an equation
  1. Step 1: Write the linear function in the form y = ax + b.
  2. Step 2: If a > 0, the quantity increases as x increases — this is linear growth.
  3. Step 3: If a < 0, the quantity decreases as x increases — this is linear decay.
  4. Step 4: The absolute value of 'a' gives the constant amount of increase or decrease per unit change in x.
💡 Students often confuse growth and decay with the sign of the y-intercept (b). The sign of the slope (a) alone determines growth (positive) or decay (negative).
Exam Traps — Don't Make These Mistakes
⚠ Confusing coefficient with constant term
✗ In the expression 5y³ + y² + 2y – 1, students write 'the coefficient of y is 1' or 'the constant term is 1'.
→ The coefficient of y is 2 (the number multiplying y), and the constant term is –1 (the term without a variable). The sign belongs to the term.
✓ Coefficient of y is 2; constant term is –1.
⚠ Degree of a constant polynomial
✗ For the polynomial –9, students write 'degree is 1' or 'degree is undefined'.
→ A constant polynomial can be written as –9x⁰, so the highest power of the variable is 0, hence degree is 0.
✓ The degree of –9 is 0.
⚠ Sign error when substituting negative values
✗ For the linear polynomial 2x + 3, when x = –6, students write '2 × –6 + 3 = –12 + 3 = –9' but sometimes they write '2 × 6 + 3 = 15' or '2 × –6 + 3 = –12 – 3 = –15'.
→ They forget that multiplying a positive by a negative gives a negative, and adding a positive to a negative follows integer rules.
✓ 2 × (–6) + 3 = –12 + 3 = –9.
⚠ Mistaking slope for y-intercept
✗ For the equation y = 2x + 5, students write 'slope is 5, y-intercept is 2'.
→ In y = ax + b, the coefficient of x (a) is the slope, and the constant term (b) is the y-intercept. Here a = 2, b = 5.
✓ Slope is 2, y-intercept is 5.
⚠ Incorrectly solving for the linear relationship from two points
✗ Given (10, 350) and (20, 550), students write '350 = 10a + b and 550 = 20a + b, then subtract to get 200 = 10a, so a = 20, then b = 350 – 200 = 150' but sometimes they add instead of subtract, getting a = 30.
→ Subtracting the first equation from the second gives (550 – 350) = (20a – 10a) + (b – b), so 200 = 10a. Adding would give 900 = 30a + 2b, which is incorrect.
✓ 550 – 350 = (20a + b) – (10a + b) → 200 = 10a → a = 20. Then b = 350 – 10(20) = 150.
⚠ Forgetting to include the constant difference in linear patterns
✗ For the pattern 1, 3, 5, 7, …, students write the nth term as '2n' instead of '2n – 1'.
→ The first term (n=1) should give 1, but 2(1) = 2, not 1. The correct expression must account for the starting value.
✓ The nth term is 2n – 1, because when n = 1, 2(1) – 1 = 1; when n = 2, 2(2) – 1 = 3, etc.
Compare & Contrast
Degree
Linear polynomial
Quadratic polynomial
A linear polynomial has degree 1 (e.g., 3z + 7), while a quadratic polynomial has degree 2 (e.g., x² + 5x + 1).
Direction of change
Linear growth
Linear decay
Linear growth occurs when a quantity increases by a constant amount over equal intervals (positive slope, e.g., C(d) = 100 + 60d), while linear decay occurs when a quantity decreases by a constant amount (negative slope, e.g., h(t) = 3 – 0.5t).
Role in the equation y = ax + b
Slope (a)
y-intercept (b)
The slope (a) determines the steepness and direction of the line (rate of change), while the y-intercept (b) determines where the line crosses the y-axis (value of y when x = 0).
Exam Focus
Marks pattern: 1-2 MCQs (1 mark each) on degree/coefficient identification, 1 SA (2-3 marks) on evaluating a polynomial or forming a linear equation, 1 LA (4-5 marks) on word problems involving linear relationships or graph-based questions.
Most asked: Identifying degree of a polynomial · Finding coefficients and constant terms · Evaluating a polynomial for a given variable value · Forming and solving linear equations from word problems · Finding linear relationships from given data points · Plotting linear equations and identifying slope and y-intercept
Don't skip: Word problems involving linear equations (e.g., age, money, perimeter problems) — typically 4-5 marks; Graph-based questions: plotting a linear equation, finding slope and y-intercept, identifying parallel lines — typically 3-5 marks; Finding the linear relationship y = ax + b from two given points — typically 3 marks
One-Line Revision (read aloud before the exam)
  1. A polynomial is an algebraic expression with one variable and non-negative integer powers.
  2. The degree of a polynomial is the highest power of the variable.
  3. A linear polynomial has degree 1, e.g., 2x + 3.
  4. A quadratic polynomial has degree 2, e.g., x² + 5x + 1.
  5. A cubic polynomial has degree 3, e.g., 5y³ + y² + 2y – 1.
  6. A constant polynomial has degree 0, e.g., –9.
  7. In y = ax + b, 'a' is the slope and 'b' is the y-intercept.
  8. Linear growth has a positive slope; linear decay has a negative slope.
  9. A linear pattern has a constant difference between consecutive terms.
  10. Parallel lines have the same slope but different y-intercepts.
  11. To evaluate a polynomial, substitute the given value of the variable and simplify.
  12. A linear equation is formed by equating a linear polynomial to a constant.
  13. The y-intercept is the point where the line cuts the y-axis at (0, b).
  14. The slope represents the constant rate of change in a linear relationship.
  15. Lines of the form y = ax pass through the origin (0, 0).

MCQs — 10 Questions with Answers

Q1. What is the degree of the polynomial x⁴ - 3x³ + 6x² - 2x + 7?

  • A. 4 ✓
  • B. 3
  • C. 2
  • D. 1

Answer: A — The degree is determined by the highest power of the variable, which is x⁴, so degree = 4.

Q2. Which of the following is a linear polynomial?

  • A. x² + 5x + 1
  • B. 3z + 7 ✓
  • C. 5y³ + y² + 2y - 1
  • D. 4x² - 3x

Answer: B — 3z + 7 has degree 1, making it a linear polynomial; the others have degrees 2, 3, and 2 respectively.

Q3. What is the constant term in the polynomial 9x³ + 5x² - 8x - 10?

  • A. -8
  • B. 5
  • C. -10 ✓
  • D. 9

Answer: C — The constant term is the term without any variable, which is -10.

Q4. A chess club charges ₹200 joining fee plus ₹50 per match. If a member played m matches, the total cost is represented by the polynomial 200 + 50m. What will be the cost for 5 matches?

  • A. ₹400
  • B. ₹450
  • C. ₹500
  • D. ₹550 ✓

Answer: D — Substituting m = 5: 200 + 50(5) = 200 + 250 = ₹550.

Q5. The perimeter of a square with side x cm is 4x cm. What is the perimeter when x = 2.5 cm?

  • A. 8 cm
  • B. 10 cm ✓
  • C. 12 cm
  • D. 15 cm

Answer: B — Perimeter = 4x = 4(2.5) = 10 cm.

Q6. In the polynomial 3x² + 5x - 2, what is the coefficient of x?

  • A. 3
  • B. 5 ✓
  • C. -2
  • D. 2

Answer: B — The coefficient of x is the number directly multiplying x, which is 5.

Q7. Which of the following is NOT a polynomial?

  • A. 2x + 3
  • B. x² + 1
  • C. 5x⁻¹ + 2 ✓
  • D. 7

Answer: C — 5x⁻¹ + 2 has a negative power of x, so it is not a polynomial (polynomials require non-negative integer powers only).

Q8. Ramesh observes that in an arithmetic sequence 5, 8, 11, 14, 17, ..., the difference between consecutive terms is always 3. Which polynomial property does this demonstrate?

  • A. Polynomial degree
  • B. Linear pattern property ✓
  • C. Coefficient of x
  • D. Constant term

Answer: B — A constant difference of 3 between successive values demonstrates the linear pattern property of polynomials.

Q9. If 2x + 10 = 64, then the value of x is:

  • A. 27 ✓
  • B. 37
  • C. 54
  • D. 32

Answer: A — Solving: 2x + 10 = 64 → 2x = 54 → x = 27.

Q10. A rectangular garden has length l metres and width w metres. If the expression for total cost of fencing is 200l + 160w + 50lw, which term makes this NOT a polynomial in one variable?

  • A. 200l
  • B. 160w
  • C. 50lw ✓
  • D. All terms are valid

Answer: C — The term 50lw contains two variables (l and w), so the entire expression is a polynomial in two variables, not one variable.

Flashcards

What is a polynomial in one variable?

An algebraic expression involving one variable and its non-negative integer powers with real coefficients.

Define the degree of a polynomial.

The highest power of the variable that appears in the polynomial with a non-zero coefficient.

What is a linear polynomial?

A polynomial of degree 1, written in the form ax + b where a ≠ 0 and a, b are real numbers.

What is the coefficient of x in the polynomial 5x² + 3x - 7?

The coefficient of x is 3.

Identify the constant term in 4z³ + 5z² - 11.

The constant term is -11.

What is a linear pattern?

A sequence where the difference between successive values at integers is constant.

What do we get when we equate a linear polynomial to a constant?

A linear equation that can be solved to find the value of the variable.

Write a linear polynomial with variable x and constant term 7.

Any polynomial of the form ax + 7 where a ≠ 0, such as 2x + 7 or 3x + 7.

What is the degree of the constant polynomial 8?

The degree is 0 because it can be written as 8x⁰.

How can we interpret a polynomial as an input-output machine?

Input the value of the variable, substitute it in the polynomial expression, and the result is the output.

Important Board Questions

Define a linear polynomial and give two examples. [2 marks]

Linear polynomial has degree 1 with form ax + b (a ≠ 0). Examples: 3x + 5, 2z - 7, etc.

The sum of two consecutive even numbers is 50. Write the linear equation and find the two numbers. [3 marks]

Let the first even number be x, then the second is x + 2. Set up equation: x + (x + 2) = 50, then solve for x.

A wire of length 30 cm is bent to form a rectangle. If one side is x cm, write the expression for the area of the rectangle and determine the value of x for which the area is maximum. Explain your reasoning. [5 marks]

Perimeter = 30, so 2(x + width) = 30, giving width = 15 - x. Area = x(15 - x) = 15x - x². This is a quadratic (not linear), but you can test values or recognize that the polynomial 15x - x² achieves maximum when x = 7.5 cm by symmetry or calculus reasoning.

Assertion-Reason Questions for Introduction to Linear Polynomials

Q1. Assertion (A):

The polynomial 3z + 7 is a linear polynomial.

Reason (R):

A linear polynomial has degree 1, which means the highest power of the variable is 1.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why 3z + 7 is linear. In this polynomial, the highest power of z is 1 (in the term 3z), making it a degree 1 polynomial, which by definition is called a linear polynomial.

Q2. Assertion (A):

In the expression 4x + 5y + 3, the coefficient of y is 5.

Reason (R):

Coefficients are the numerical values that appear in the terms of an algebraic expression regardless of the number of variables present.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true; the coefficient of y in 4x + 5y + 3 is indeed 5. However, the reason, while true, does not fully explain why this is the coefficient. The reason correctly defines coefficients but does not specifically address this particular case. A better reason would directly state that 5 is the number multiplying y in the term 5y.

Q3. Assertion (A):

A wire of length 20 cm bent to form a rectangle with length x cm has width (10 - x) cm.

Reason (R):

The perimeter of a rectangle is 2(length + width), and dividing the total wire length by 2 gives the semi-perimeter.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains the assertion. Since perimeter = 20 cm and perimeter = 2(x + width), we get 20 = 2(x + width), so x + width = 10, giving width = (10 - x) cm. The reasoning chain is logically sound and directly supports the assertion.

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Case-Based Questions (CBQ) for Introduction to Linear Polynomials

Case Study:

A rectangular garden has length l metres and width w metres. A wire fence costs ₹100 per metre and must be placed along the entire length of the garden (both sides). A wooden fence costs ₹80 per metre and must be placed along the entire width of the garden (both sides). Special seeds costing ₹50 per square metre are sown throughout the entire garden. The gardener needs to calculate the total cost before purchasing materials.

Q1. What is the cost of the wire fence for both lengths of the garden?

  • A. 100l
  • B. 200l ✓
  • C. 100lw
  • D. 50l

Answer: B

Q2. What is the cost of the wooden fence for both widths of the garden?

  • A. 80w
  • B. 160w ✓
  • C. 80lw
  • D. 100w

Answer: B

Q3. What is the cost of sowing seeds throughout the garden?

  • A. 50l + 50w
  • B. 50(l + w)
  • C. 50lw ✓
  • D. 50 + lw

Answer: C

Q4. Which term in the total cost expression 200l + 160w + 50lw depends on the area of the garden?

  • A. 200l
  • B. 160w
  • C. 50lw ✓
  • D. All terms depend on area

Answer: C

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