Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
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.
Real Numbers encompass all rational and irrational numbers, forming a complete and continuous number line essential for various mathematical concepts.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
Coordinate Geometry explores the relationship between algebra and geometry through the use of coordinate systems to represent geometric shapes and solve problems.
Explore the basics of trigonometry, including angles, triangles, and the fundamental trigonometric ratios: sine, cosine, and tangent.
Explore real-world applications of trigonometry in measuring heights, distances, and angles in various fields such as astronomy, navigation, and architecture.
Explore the properties, theorems, and applications of circles in geometry, including tangents, chords, and angles subtended by arcs.
Explore the concepts of calculating areas related to circles, including sectors, segments, and combinations with other geometric shapes.
