This chapter discusses polynomials, their degrees, and classifications such as linear, quadratic, and cubic. Understanding polynomials is essential for solving various mathematical problems.
Polynomials – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Polynomials chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Linear Polynomial: p(x) = ax + b
a and b are coefficients, a ≠ 0. Represents a straight line when graphed. Example: 2x + 3.
Quadratic Polynomial: p(x) = ax² + bx + c
a, b, c are coefficients, a ≠ 0. Graphs as a parabola. Example: x² - 3x - 4.
Cubic Polynomial: p(x) = ax³ + bx² + cx + d
a, b, c, d are coefficients, a ≠ 0. Can have up to 3 zeroes. Example: 2x³ - 5x² - 14x + 8.
Zero of a Polynomial: p(k) = 0
k is a zero of p(x) if substituting x with k yields 0. Example: For p(x) = x² - 3x - 4, p(4) = 0.
Sum of Zeroes (Quadratic): α + β = -b/a
α, β are zeroes. Relates sum of zeroes to coefficients. Example: For x² + 7x + 10, sum is -7.
Product of Zeroes (Quadratic): αβ = c/a
Relates product of zeroes to coefficients. Example: For x² + 7x + 10, product is 10.
Sum of Zeroes (Cubic): α + β + γ = -b/a
α, β, γ are zeroes. Example: For 2x³ - 5x² - 14x + 8, sum is 5/2.
Sum of Products of Zeroes (Cubic): αβ + βγ + γα = c/a
Example: For 2x³ - 5x² - 14x + 8, sum of products is -7.
Product of Zeroes (Cubic): αβγ = -d/a
Example: For 2x³ - 5x² - 14x + 8, product is -4.
Degree of a Polynomial
Highest power of x in p(x). Determines maximum number of zeroes. Example: x³ has degree 3.
Equations
Finding Zeroes of Linear Polynomial: ax + b = 0 ⇒ x = -b/a
Solution gives the zero of the polynomial. Example: 2x + 3 = 0 ⇒ x = -3/2.
Finding Zeroes of Quadratic Polynomial: ax² + bx + c = 0
Use factorization or quadratic formula. Example: x² - 3x - 4 = 0 ⇒ x = -1, 4.
Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a
Directly finds zeroes of any quadratic polynomial. Example: For x² - 3x - 4, x = [3 ± √(9 + 16)] / 2.
Factor Theorem: If p(k) = 0, then (x - k) is a factor of p(x)
Useful for factorizing polynomials. Example: For p(x) = x² - 3x - 4, p(4) = 0 ⇒ (x - 4) is a factor.
Remainder Theorem: Remainder when p(x) is divided by (x - a) is p(a)
Quickly find remainders without division. Example: p(x) = x³ - 2x² + x - 1 divided by (x - 2) gives p(2) = 1.
Identity for Zeroes: p(x) = k(x - α)(x - β) for quadratic
Expresses polynomial in terms of its zeroes. Example: p(x) = x² - 5x + 6 = (x - 2)(x - 3).
Identity for Zeroes (Cubic): p(x) = k(x - α)(x - β)(x - γ)
Example: p(x) = x³ - 6x² + 11x - 6 = (x - 1)(x - 2)(x - 3).
Relation Between Coefficients and Zeroes (Quadratic): α + β = -b/a, αβ = c/a
Connects coefficients directly to zeroes. Example: For x² - 5x + 6, α + β = 5, αβ = 6.
Relation Between Coefficients and Zeroes (Cubic): α + β + γ = -b/a, αβ + βγ + γα = c/a, αβγ = -d/a
Extends quadratic relations to cubic polynomials. Example: For 2x³ - 5x² - 14x + 8, relations hold as shown.
Graphical Representation: y = p(x) intersects x-axis at zeroes
Visual method to find zeroes. Example: y = x² - 3x - 4 intersects x-axis at x = -1 and x = 4.
This chapter explores real numbers, focusing on key properties such as the Fundamental Theorem of Arithmetic and the concept of irrational numbers, which are crucial for understanding the number system.
This chapter focuses on solving pairs of linear equations with two variables and their real-life applications.
This chapter explores quadratic equations, highlighting their forms and significance in real-world applications.
This chapter introduces arithmetic progressions, which are sequences of numbers generated by adding a fixed value to the previous term. Understanding these patterns is crucial for solving real-life mathematical problems.
This chapter focuses on the properties of triangles, specifically their similarity and how it can be applied in various real-world contexts.
This chapter covers the concepts of coordinate geometry, including finding distances between points and dividing line segments. Understanding these concepts is essential for solving geometry problems using algebra.
This chapter focuses on the foundational concepts of trigonometry, particularly the relationships between the angles and sides of right triangles.
This chapter explores how trigonometry is applied in real-life situations, particularly in measuring heights and distances.
This chapter explores the properties of circles, particularly focusing on tangents and their relationship with radii and secants.
This chapter focuses on sectors and segments of circles, essential concepts in geometry. Understanding these helps in solving real-life problems related to areas and measurements.
