Table of contents | |
Introduction | |
Polynomials in One Variable | |
Zeroes of a Polynomial | |
Factorization of Polynomials | |
Algebraic Identities |
In mathematics, a variable is denoted by a symbol that can take any real value, often represented by letters such as x, y, z, etc. Expressions like 2x, 3x, -x, and -1/2x are examples of algebraic expressions, specifically in the form (a constant) × x. When the constant is unknown, it is denoted as a, b, c, etc.
A polynomial is an algebraic expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication operations.
“The expression which contains one or more terms with non – zero coefficient is called polynomial”
Polynomials are algebraic expressions with variables, coefficients, and exponents. When the exponents are whole numbers, the expressions are termed polynomials in one variable.
Example: x³ - x² + 4x + 7 and 3y² + 5y.
Terms and CoefficientsIn a polynomial like x² + 2x, x² and 2x are referred to as terms. Each term has a coefficient—in -x³ + 4x² + 7x - 2, coefficients are -1, 4, 7, and -2. The term x⁰ (where x⁰ = 1) is also present.
Constant Polynomials and Zero Polynomials
Constants like 2, -5, and 7 are examples of constant polynomials. The constant polynomial 0 is termed the zero polynomial, a significant concept in polynomial theory.
Expressions like x + 2/x, x⁻¹, and x + 3√(x) aren't polynomials due to non-whole number exponents.
Polynomials can be denoted by symbols like p(x), q(x), or r(x), where the variable is x. Examples include:
The degree of a polynomial is the highest power of its variable. For example, in 3x⁷ - 4x⁶ + x + 9, the degree is 7. Constant polynomials have a degree of 0.
Examples:
A polynomial in one variable of degree n is written as: anx^{n}+an−1x^{n-1}+…+a^{1}x+a^{0} where a^{0},a^{1},…, a^{n} are constants and a^{n}=0.
The zero polynomial, denoted as 0, has an undefined degree. Polynomials can extend to more than one variable, like x^{2}+y^{2}+xyz in three variables.
Consider the polynomial p(x)=5x^{3}−2x^{2}+3x−2. To find the value of p(x) at different points, substitute the given values for x.
For Example,
Given Polynomial: p(x) = 5x^{3} - 2x^{2} + 3x - 2
Example Calculation:
1. For x = 1:
p(1) = 5 - 2 + 3 - 2 = 4
2. For x = 0:
p(0) = 0 - 0 + 0 - 2 = -2
3. For x = -1:
p(-1) = -5 - 2 - 3 - 2 = -12
In summary, for the given polynomial p(x)=5x^{3}−2x^{2}+3x−2:
These values are found by substituting the respective values of x into the polynomial expression.
Example: Value of Polynomials at Given Points
(i) For p(x)=5x^{2}−3x+7 at x=1: p(1)=5(1)^{2}−3(1)+7=9
(ii) For q(y)=3y^{3}−4y+11 at y=2: q(2)=3(2)^{3}−4(2)+11=27−8+11=30
(iii) For p(t)=4t^{4}+5t^{3}−t^{2}+6 at t=a: p(a)=4a^{4}+5a^{3}−a^{2}+6
When evaluating p(x