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Zero polynomial is a type of polynomial where all variable's coefficients are equal to zero. In other words, it means that all the variables have a power that is equal to zero. A polynomial is an expression consisting of coefficients and variables. Let us learn more about zero polynomials, the degree of zero polynomial, and solve a few examples.
|1.||What is Zero Polynomial?|
|2.||Degree of Zero Polynomial|
|3.||Zero of Zero Polynomial|
|4.||FAQs on Zero Polynomial|
What is Zero Polynomial?
Zero polynomial is a type of polynomial where the coefficients are zero and are usually written as 0 and have no terms. Zero polynomial is the only kind of polynomial that has an undefined degree. However, some mathematics define the degree of zero polynomial as negative usually written as -1 or -.
Definition of Zero Polynomial
Any polynomial with all the variables that have their coefficients equal to zero is called zero polynomial. Hence, the value of a zero polynomial is zero. The function that defines it is called a constant function or zero map usually expressed as P(x) = 0, where x is the variable of the polynomial whose coefficient is zero. A zero polynomial can have an infinite number of terms along with variables of different powers where the variables have zero as their coefficient. For example: 0x2 + 0x + 0.
The zero polynomial function is defined as y = P(x) = 0 and the graph of zero polynomial is the x-axis. The domain is considered as real numbers and the range is zero. The domain is the set of values of the variable x for which the function is defined and the range is the set of values of the variable y that is dependent.
Degree of Zero Polynomial
The degree of zero polynomial is usually undefined unless a degree is assigned then it is -1 or ∞. A degree of a polynomial is considered as the maximum degree of its non-zero terms while a zero polynomial does not have any non-zero terms. Hence, there are no terms with degrees for us to calculate the degree of a polynomial. Any non-zero number or a constant is said to be a zero degree polynomial if f(x) = a as f(x) = ax0 where a ≠ 0. For example: f(x) = 0, g(x) = 0x , h(x) = 0x2.
Zero of Zero Polynomial
Zero of zero polynomial is any number that can be a rational number, irrational number, or complex number. Since zero of a polynomial is the number that by substituting the variable results in the polynomial's value being zero. While in zero polynomial the coefficient of every term is zero. Therefore, even after substituting, the value of the polynomial will always be zero. Hence, zero itself is the zero polynomial.
Listed below are a few topics related to zero polynomial, take a look.
Examples on Zero Polynomial
Example 1: Eve knows that the zeros of a quadratic polynomial are -4 and 6. How can we help to find the equation of the polynomial?
The zeros of the quadratic polynomial are -4 and 6.
Let α = -4, and b = 6
Then, we have the sum of the roots = α + b = 2
Product of the roots = α.b = -24
The required quadratic equation is x2 - (α + b) + α.b = 0
x2 - 2(x) + (-24) = 0
x2 - 2x - 24 = 0
Therefore the equation of the quadratic polynomial is x2 - 2x - 15 = 0.
Example 2: Find the degree of the polynomial 5x4 + 3x2 - 7x5 + x7
In order to find the degree of the given polynomial, Check each term of the given polynomial. All are like terms with x as a variable. Arrange these terms in descending order of their powers, which gives x7 - 7x5 + 5x4+ 3x2 Term with the greatest or highest exponent is x7, so the degree of the polynomial is 7. Therefore, the degree of the polynomial is 7.
FAQs on Zero Polynomial
What is Zero Polynomial With Example?
A zero polynomial is a type of polynomial where the coefficients of the variables are equal to 0. The constant polynomial f(x) = 0. The general form is g(x) = ax + b where a ≠ 0. For example, f(x) = x -4, g(x) = 14x, etc. The general form is also expressed as a linear polynomial.
What are Zero Polynomial and Constant Polynomial?
A constant polynomial has its coefficients equal to 0. Whereas a zero polynomial is the additive identity of the additive groups of polynomials such as f(x) = 0. In a constant polynomial, the degree is 0 whereas in a zero polynomial, the degree is undefined or written as -1.
How Many Zeros Does a Zero Polynomial Have?
In a zero polynomial, the coefficients equal to 0 i.e. f(x) = 0, where x is any value. Hence, the result of f(x) will be 0. Therefore, the number of zeros in a zero polynomial is infinite.
What is the Degree of Zero Polynomial?
The degree of a zero polynomial is either undefined or is expressed as -1. Just as any constant value, 0 can be considered as a constant value known as zero polynomial.
What is Zero of Zero Polynomial?
zero polynomial is considered as a constant polynomial with all the coefficients equal to 0. Whereas zero of a polynomial is the value of the variable that makes the polynomial equal to 0. Therefore, zero of zero polynomial is any real number.