We have sent an OTP to your contact. Please enter it below to verify.
Alert
Your message here...
Your notification message here...
For any questions or assistance regarding Customer Support, Sales Inquiries, Technical Support, or General Inquiries, our AI-powered team is here to help!
This section explains the significance of zeroes in polynomials through their geometrical representation. It covers how the graphs of linear and quadratic polynomials intersect the x-axis and describes the cases where zeroes can be found in those graphs.
The zeroes of a polynomial are the x-values where the polynomial intersects the x-axis, represented as real numbers k such that p(k) = 0. The section examines the importance of these zeroes through their geometrical interpretations in linear and quadratic polynomial graphs.
The number of zeroes corresponds to the degree of the polynomial, with quadratics having at most two and cubics having at most three. This relationship between the degree of the polynomial and its zeroes is a foundational concept in understanding polynomial behavior.
Zero of a Polynomial: Refers to the values of x where the polynomial intersects the x-axis.
Linear Polynomials: Have one zero, represented by the formula -b/a.
Quadratic Polynomials: Can have up to two zeroes based on their graph shape (two, one, or none).
Cubic Polynomials: Can have up to three zeroes, depending on how many times they intersect the x-axis.
A quadratic so fair, intersects with care; with shapes to show, two, one, or none, let's go!
Imagine a racecar (the polynomial) trying to reach two checkpoints (zeroes) at a mountain slope (the graph). Sometimes, it reaches both, sometimes only one, or maybe none at all!
P.O.S. for Polynomial Observations: Points Of intersection are the zeroes!
Example 1: For the polynomial y = 2x + 3, the zero is x = -3/2.
Example 2: The quadratic polynomial y = x² - 3x - 4 has zeroes at x = -1 and x = 4.
Example 3: The cubic polynomial y = x³ - 4x has zeroes at x = -2, x = 0, and x = 2.
Term: Polynomial
Definition: An algebraic expression formed by the sum of powers in one or more variables multiplied by coefficients.
An algebraic expression formed by the sum of powers in one or more variables multiplied by coefficients.
Term: Zero of a Polynomial
Definition: A real number k such that p(k) = 0, indicating where the polynomial intersects the x-axis.
A real number k such that p(k) = 0, indicating where the polynomial intersects the x-axis.
Term: Linear Polynomial
Definition: A polynomial of degree one, which is represented as y = ax + b.
A polynomial of degree one, which is represented as y = ax + b.
Term: Quadratic Polynomial
Definition: A polynomial of degree two, expressed as y = ax² + bx + c.
A polynomial of degree two, expressed as y = ax² + bx + c.
Term: Cubic Polynomial
Definition: A polynomial of degree three, represented as y = ax³ + bx² + cx + d.
A polynomial of degree three, represented as y = ax³ + bx² + cx + d.
Term: Graph
Definition: A visual representation of the relationship between two variables, often plotted in a coordinate system.
A visual representation of the relationship between two variables, often plotted in a coordinate system.
Term: Xaxis
Definition: The horizontal axis in a coordinate plane where y=0.
The horizontal axis in a coordinate plane where y=0.
Term: Intersection
Definition: The points where the graph of a function crosses the x-axis.
The points where the graph of a function crosses the x-axis.
Term: Parabola
Definition: The U-shaped graph of a quadratic function.
The U-shaped graph of a quadratic function.