Zeros of Polynomial
Zeros of a polynomial, also known as roots or solutions, are the values of the variable that make the polynomial equal to zero. In other words, they are the values of x for which the polynomial evaluates to zero.
How to Find Zeros of Polynomials?
The number of zeros that a polynomial contains is determined by the degree of the polynomial equation. Polynomial equations are categorized into various types such as linear equations, quadratic equations, cubic equations, and higher-degree polynomials. Each type of equation is examined separately in order to determine the zeros of the polynomial.
Linear Polynomial
A linear polynomial is a polynomial with degree 1. The standard form is y = ax + b, where a and b are real numbers and a≠0.
Examples of linear polynomials: 2x + 3, a – 6b, y – 12, etc.
Zeroes of a Linear Polynomial
The zero of this equation can be calculated by substituting y = 0, and on simplification, we have ax + b = 0, or x = -b/a.
Quadratic Polynomial
A polynomial of degree 2 is known as a quadratic polynomial. The standard form is ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0.
Examples of Quadratic Polynomials: x2+ 3x + 4, 3y2+ 7xy + 4y, etc.
Zeroes of Quadratic polynomial
Quadratic equation of the form x2 + x(a + b) + ab = 0 can be factorized as
(x + a)(x + b) = 0,
Where we have x = -a, and x = -b as the zeros of the polynomial.
And for a quadratic equation of the form ax2+ bx + c = 0, which cannot be factorized, the zeros can be calculated using the formula method, and the formula is
x = [- b ± √(b2 – 2ac) ] / 2a
Cubic Polynomial
A cubic polynomial is a polynomial of degree three. The standard form is ax3+ bx2 + cx + d, where a, b, c, and d are real numbers and a≠0.
Examples of a Cubic Polynomials: x3 + 4x2 + 7x + 2, 3y3 – 2y2+ 4y – 7, etc.
Zeroes of Cubic Polynomial
The cubic equation of the form y = ax3 + bx2 + cx + d, can be factorized by applying the remainder theorem. According to the remainder theorem, if we substitute a smaller value, denoted as α, for the variable x and the resulting value of y is zero (y = 0), then (x – α) is one of the roots of the equation. By dividing the cubic equation by (x – α) using long division, we can obtain a quadratic equation. The quadratic equation can then be solved either through factorization or by using the formula method to find the two desired roots of the equation.
Read More about Solving Cubic Equations.
Polynomials – Definition, Standard Form, Types, Identities, Zeroes
Polynomials: In mathematics, polynomials are mathematical expressions consisting of indeterminates (also called variables) and coefficients, that involve only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. They are used in various fields of mathematics, astronomy, economics, etc. There are various examples of the polynomials such as 2x + 3, x2 + 4x + 5, etc.
In this article, we will learn about, Polynomials, Degrees of Polynomials, Examples of Polynomials, Zeros of Polynomials, Polynomial Equations, and others in detail.
Table of Content
- What are Polynomials?
- Polynomials Definition
- Polynomials Examples
- Characteristics of Polynomials
- Standard Form of a Polynomial
- Degree of a Polynomial
- Degree of Single Variable Polynomial
- Degree of a Multivariable Polynomial
- Terms in a Polynomial
- Types of Polynomials
- Properties of Polynomials (Theorems of Polynomials)
- Operations on Polynomials
- Addition of Polynomials
- Subtraction of Polynomials
- Multiplication of Polynomials
- Division of Polynomials
- Factorization of Polynomials
- Methods of Factorization of Polynomial
- Greatest Common Factor (GCF)
- Substitution Method
- Grouping Method
- Difference of Two Squares Identity
- Zeros of Polynomial
- How to Find Zeros of Polynomials?
- Linear Polynomial
- Quadratic Polynomial
- Cubic Polynomial
- Higher Degree Polynomial
- Polynomial Identities
- Polynomial Equations
- Solving Polynomials
- Polynomial Functions
- Polynomials Class 9 Extra Questions
- Polynomials Class 10 Extra Questions
- Practice Problems on Polynomial