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.
