Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Polynomials - Quick Look Revision Guide
Your 1-page summary of the most exam-relevant takeaways from Mathematics.
This compact guide covers 20 must-know concepts from Polynomials aligned with Class X preparation for Mathematics. Ideal for last-minute revision or daily review.
Complete study summary
Essential formulas, key terms, and important concepts for quick reference and revision.
Key Points
Define polynomial with an example.
A polynomial is an expression of the form ax^n + bx^(n-1) + ... + k, where a, b, ..., k are constants, and n is a non-negative integer. Example: 2x^2 + 3x + 1.
Degree of a polynomial.
The highest power of x in a polynomial is called its degree. For example, in 4x^3 + 2x^2 + x + 7, the degree is 3.
Linear polynomial definition.
A polynomial of degree 1 is called a linear polynomial. Example: 3x + 5.
Quadratic polynomial definition.
A polynomial of degree 2 is called a quadratic polynomial. Example: x^2 - 5x + 6.
Cubic polynomial definition.
A polynomial of degree 3 is called a cubic polynomial. Example: 2x^3 - x^2 + 4x - 8.
Zeroes of a polynomial.
The values of x for which the polynomial p(x) becomes zero are called zeroes of the polynomial. For p(x) = x^2 - 3x - 4, zeroes are 4 and -1.
Geometrical meaning of zeroes.
Zeroes of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
Graph of a linear polynomial.
The graph of a linear polynomial ax + b is a straight line intersecting the x-axis at (-b/a, 0).
Graph of a quadratic polynomial.
The graph of a quadratic polynomial ax^2 + bx + c is a parabola. It can intersect the x-axis at two, one, or no points.
Sum of zeroes of quadratic polynomial.
For ax^2 + bx + c, sum of zeroes (α + β) = -b/a.
Product of zeroes of quadratic polynomial.
For ax^2 + bx + c, product of zeroes (αβ) = c/a.
Formation of quadratic polynomial.
A quadratic polynomial with zeroes α and β can be written as x^2 - (α + β)x + αβ.
Graph of a cubic polynomial.
The graph of a cubic polynomial can intersect the x-axis at most three times, corresponding to its zeroes.
Sum of zeroes of cubic polynomial.
For ax^3 + bx^2 + cx + d, sum of zeroes (α + β + γ) = -b/a.
Sum of product of zeroes taken two at a time.
For ax^3 + bx^2 + cx + d, αβ + βγ + γα = c/a.
Product of zeroes of cubic polynomial.
For ax^3 + bx^2 + cx + d, product of zeroes (αβγ) = -d/a.
Division algorithm for polynomials.
Given polynomials p(x) and g(x), there exist q(x) and r(x) such that p(x) = g(x) * q(x) + r(x), where degree of r(x) < degree of g(x).
Finding zeroes using factor theorem.
If p(a) = 0, then (x - a) is a factor of p(x), and a is a zero of p(x).
Important identity: a^2 - b^2.
a^2 - b^2 = (a - b)(a + b). Useful for factoring polynomials.
Misconception: All expressions are polynomials.
Expressions like 1/(x - 1) or √x are not polynomials as they violate polynomial definitions.
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.
