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.
