CH – 2 POLYNOMIALS
Class 10 Mathematics — CBSE Board Revision
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Chapter Overview
In this chapter, students will explore algebraic expressions known as polynomials in one variable. The chapter covers:
- Definition, terms, coefficients, and degree of polynomials.
- Classification of polynomials based on their degree (linear, quadratic, cubic).
- Definition of zeroes of a polynomial and their calculation.
- Geometrical representation of zeroes as x-coordinates where the graph intersects the x-axis.
- Relationship between zeroes and coefficients of a quadratic polynomial.
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🎯 BOARD EXAM FOCUS
- High-priority concepts to practice: Finding zeroes of quadratic polynomials by factorisation and verifying their relationship with coefficients.
- Important skills for exam preparation: Determining the number of zeroes of a polynomial directly from its graph.
- Concepts students should revise carefully: Forming a quadratic polynomial when the sum and product of zeroes are given.
- Essential practice areas: Splitting the middle term correctly and maintaining sign accuracy during algebraic transformations.
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What is a Polynomial?
A polynomial p(x) in one variable x is an algebraic expression composed of variables, exponents, and coefficients, where the exponent of the variable x is always a non-negative integer.
- Terms: The parts of the expression separated by ‘+’ or ‘-‘ operators.
- Coefficients: The real numbers multiplied by the variable in each term.
- Degree: The highest power/exponent of the variable x in p(x).
Examples:
- p(x) = 4x + 2 is a polynomial in variable x of degree 1.
- p(y) = 2y² – 3y + 4 is a polynomial in variable y of degree 2.
- p(x) = 5x³ – 4x² + x – 2 is a polynomial of degree 3.
Non-Examples:
- 1/(x – 1), √(x) + 2, and 1/(x² + 2x + 3) are NOT polynomials because the variable exponents are negative or fractional.
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Types of Polynomials
Polynomials are classified according to their degree:
- Linear Polynomial
- Degree = 1
- General form: ax + b (where a ≠ 0)
- Examples: 2x – 3, √3 x + 5, 3z + 4
- Quadratic Polynomial
- Degree = 2
- General form: ax² + bx + c (where a ≠ 0 and a, b, c are real numbers)
- Examples: x² – 3x – 4, 2x² + 5x – 3, y² – 2
- Cubic Polynomial
- Degree = 3
- General form: ax³ + bx² + cx + d (where a ≠ 0 and a, b, c, d are real numbers)
- Examples: x³ – 4x, 2 – x³, 3x³ – 2x² + x – 1
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Zeroes of a Polynomial
A real number k is called a zero of a polynomial p(x) if p(k) = 0.
Finding Zeroes:
- For a linear polynomial p(x) = ax + b, set p(x) = 0 ⟹ ax + b = 0 ⟹ x = -b/a.
- For a quadratic polynomial, set p(x) = 0 and solve by splitting the middle term to obtain factors.
Example:
Consider p(x) = x² – 3x – 4.
p(-1) = (-1)² – 3(-1) – 4 = 1 + 3 – 4 = 0
p(4) = (4)² – 3(4) – 4 = 16 – 12 – 4 = 0
Since p(-1) = 0 and p(4) = 0, the real numbers -1 and 4 are zeroes of p(x).
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Geometrical Meaning of Zeroes
Geometrically, the zeroes of a polynomial p(x) are precisely the x-coordinates of the points where the graph of y = p(x) intersects or touches the x-axis.
- Linear Polynomial (y = ax + b): The graph is a straight line that intersects the x-axis at exactly one point (-b/a, 0).
- Quadratic Polynomial (y = ax² + bx + c): The graph is a U-shaped curve called a parabola. It opens upwards if a > 0 and downwards if a < 0.
- Case 1: Intersects x-axis at two distinct points ⟹ 2 zeroes.
- Case 2: Touches x-axis at exactly one point ⟹ 1 zero (two coincident zeroes).
- Case 3: Does not intersect x-axis ⟹ 0 real zeroes.
- A polynomial p(x) of degree n intersects the x-axis at at most n points, meaning it has at most n zeroes.
⭐ REMEMBER
The total number of zeroes of a polynomial y = p(x) is equal to the total number of points where its graph intersects or touches the X-AXIS. Do not count intersections with the y-axis!
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Relationship Between Zeroes and Coefficients
For a Quadratic Polynomial p(x) = ax² + bx + c (where a ≠ 0):
If α (alpha) and β (beta) are the zeroes of p(x), then (x – α) and (x – β) are its factors.
- Sum of Zeroes:
α + β = -b / a = -(Coefficient of x) / (Coefficient of x²) - Product of Zeroes:
α × β = c / a = (Constant term) / (Coefficient of x²)
Worked Example:
Find the zeroes of p(x) = x² + 7x + 10 and verify the relationship with its coefficients.
Factorisation:
x² + 7x + 10 = x² + 2x + 5x + 10 = x(x + 2) + 5(x + 2) = (x + 2)(x + 5)
Set p(x) = 0 ⟹ x = -2 or x = -5.
So, zeroes are α = -2 and β = -5.
Verification:
- Sum of zeroes: α + β = (-2) + (-5) = -7 = -7 / 1 = -(Coefficient of x) / (Coefficient of x²)
- Product of zeroes: α × β = (-2) × (-5) = 10 = 10 / 1 = (Constant term) / (Coefficient of x²)
Forming a Quadratic Polynomial:
If sum (S) and product (P) of zeroes are given, the quadratic polynomial is:
k[x² – (α + β)x + αβ] = k[x² – Sx + P], where k is any non-zero real number.
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📊 QUICK RELATIONSHIP SUMMARY
- Linear Polynomial: ax + b
- Zero = -b/a = -(Constant term) / (Coefficient of x)
- Quadratic Polynomial: ax² + bx + c
- Sum of zeroes (α + β) = -b/a
- Product of zeroes (α × β) = c/a
- Cubic Polynomial: ax³ + bx² + cx + d
- Sum of zeroes (α + β + γ) = -b/a
- Sum of product of zeroes taken two at a time (αβ + βγ + γα) = c/a
- Product of zeroes (αβγ) = -d/a
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Division Algorithm for Polynomials
If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that:
p(x) = g(x) × q(x) + r(x)
where r(x) = 0 or degree of r(x) < degree of g(x).
- p(x) is Dividend
- g(x) is Divisor
- q(x) is Quotient
- r(x) is Remainder
Usage:
- To divide a higher degree polynomial by a lower degree polynomial.
- To check if a given polynomial g(x) is a factor of p(x) (if remainder r(x) = 0, then g(x) is a factor).
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🧠 Concepts Students Find Difficult
- Graphical Intersections: Students often mistakenly count the point where the curve crosses the y-axis as a zero. Zeroes depend ONLY on x-axis intersections.
- Negative Signs in Relationships: Confusing α + β = -b/a with positive b/a. Remember the negative sign belongs to the coefficient of x!
- Re-arranging Terms Before Factorising: For polynomials like 6x² – 3 – 7x, students forget to rearrange into standard form 6x² – 7x – 3 before splitting the middle term.
- Creating Polynomial from Zeroes: Forgetting the minus sign in k[x² – (Sum)x + Product] when given the sum and product directly.
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⚠️ Common Mistakes Students Make
- Misidentifying Non-Polynomial Expressions
- Mistake: Treating 2x² + 3√x – 5 as a polynomial.
- How to avoid: Ensure all variable exponents are non-negative whole numbers (0, 1, 2, …). Square roots on variables mean fractional powers, which are not allowed.
- Sign Errors in -b/a Formula
- Mistake: For 2x² – 8x + 6, writing Sum = 8/2 = 4, but getting confused if b is already negative.
- How to avoid: Always write -b/a as -(-8)/2 = 8/2 = 4. Use parentheses for negative values.
- Counting Y-Axis Intersections
- Mistake: Stating a graph has 3 zeroes because it touches the x-axis twice and crosses the y-axis once.
- How to avoid: Ignore the y-axis completely when finding zeroes graphically. Focus solely on x-coordinates.
- Missing Terms When Rearranging Standard Form
- Mistake: Taking b = -3 and c = -7 directly from 6x² – 3 – 7x.
- How to avoid: Always rewrite the polynomial in decreasing powers of x first: 6x² – 7x – 3. Here, a = 6, b = -7, c = -3.
- Forgetting Constant ‘k’ in Quadratic Polynomial Formation
- Mistake: Writing only x² – (1/4)x – 1 without clearing fractions.
- How to avoid: Multiply by a constant k to clear denominators: k(x² – (1/4)x – 1) = 4x² – x – 4 when k = 4.
- Incomplete Middle-Term Splitting
- Mistake: Stopping at factorisation and forgetting to solve for x = 0 to get the zeroes.
- How to avoid: Set each factor to zero explicitly: x + 2 = 0 ⟹ x = -2.
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🔥 Priority Revision
⭐⭐⭐ HIGH-PRIORITY PRACTICE
- Finding zeroes of quadratic polynomials using factorisation and verifying relations α + β = -b/a, αβ = c/a.
- Finding the quadratic polynomial given the sum and product of its zeroes.
- Identifying the number of zeroes from given graphs of y = p(x).
⭐⭐ IMPORTANT PRACTICE
- Factorising polynomials with irrational coefficients or missing terms (e.g., t² – 15, 4u² + 8u).
- Checking whether a given expression is a factor of another polynomial.
⭐ ADDITIONAL REVISION
- Basic division of polynomials and verifying Dividend = Divisor × Quotient + Remainder.
- Conceptual properties of cubic polynomial zeroes.
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⚡ Polynomials Quick Revision Sheet
- Degree of Polynomial: Highest power of x in p(x).
- Linear: Degree 1 | Form: ax + b | Max Zeroes: 1
- Quadratic: Degree 2 | Form: ax² + bx + c | Max Zeroes: 2
- Cubic: Degree 3 | Form: ax³ + bx² + cx + d | Max Zeroes: 3
- Geometrical Zeroes: Number of points where y = p(x) crosses/touches the X-AXIS.
- Quadratic Relations: α + β = -b/a, αβ = c/a
- Formation Formula: p(x) = k[x² – (Sum)x + Product]
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📝 Test Yourself
- Find the degree of the polynomial p(x) = 3x³ – 5x² + 4x – 7.
- If the graph of y = p(x) intersects the x-axis at 3 distinct points and touches it at 1 point, how many zeroes does p(x) have?
- Find the zeroes of the quadratic polynomial x² – 5x + 6 and verify the relationship between zeroes and coefficients.
- Find the zeroes of the polynomial t² – 12 and verify the relationship with its coefficients.
- Find a quadratic polynomial whose sum and product of zeroes are -1/3 and 2/3 respectively.
- If α and β are the zeroes of the polynomial 2x² – 7x + 3, find the value of (α + β) + αβ.
- Check whether the expression x + 3 is a factor of the polynomial x³ + 3x² – 2x – 6 using polynomial division.
- If one zero of the quadratic polynomial x² + 3x + k is 2, find the value of k.
- Divide 2x² + 3x + 1 by x + 2 and find the quotient and remainder.
- Can a quadratic polynomial have no real zeroes? Explain geometrically.
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🎯 Before Your Exam
☐ I understand what a polynomial is and how to determine its degree.
☐ I can identify linear, quadratic, and cubic polynomials.
☐ I can find the number of zeroes of y = p(x) from a graph.
☐ I can factorise quadratic polynomials by splitting the middle term.
☐ I know how to calculate zeroes for polynomials like x² – a² or ax² + bx.
☐ I know how to verify α + β = -b/a and αβ = c/a.
☐ I can form a quadratic polynomial when given the sum and product of zeroes.
☐ I understand the basic division algorithm for polynomials.
Believe in your preparation, stay calm, and write your answers neatly. Success is yours!