Zeroes of Polynomial - MathsTips.com

A polynomial having value zero (0) is known as zero polynomial. Actually, the term 0 is itself zero polynomial. It is a constant polynomial whose all the coefficients are equal to 0. For a polynomial, there may be few (one or more) values of the variable for which the polynomial may result in zero. These values are known as zeros of a polynomial. We can say that the zeroes of a polynomial are defined as the points where the polynomial equals to zero on the whole.

If the coefficients of following the form of the polynomial: $a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+....+a_{2}x^{2}+a_{1}x+a_{0}$ are zero, then it will become zero polynomial. i.e $a_{n}=a_{n-1}=a_{n-2}=...=a_{0}=0$ . Thus, the polynomial will become 0 and may be written as P(x)=0.

Zero polynomial function

The zero polynomial function is defined as the polynomial function with the value of zero. i.e. the function whose value is 0, is termed as a zero polynomial function. Zero polynomial does not have any nonzero term. It is represented as: P(x) = 0. Thus, we can say that a polynomial function which is equal to zero, is called zero polynomial function. It also is known as zero map. The graph of the zero polynomial is X axis.

Zero quadratic polynomial

The quadratic polynomial having all the coefficients equal to zero is known as zero quadratic polynomial. The general term of a quadratic polynomial is: $P(x)=ax^{2}+bx+c$ . If in above quadratic polynomial, the coefficients are zero; i.e. a = b = c = 0, then the polynomial is termed as a zero quadratic polynomial.

Example 1: $0.x^{2}+0.x+0$ is a zero quadractic polynomial whose values are zero.

Example 2: Find the additive identities of the following polynomials: 1) x-3 and 2) $x^{2}-3x+5$

Solution: 1) Additive identity = 0.x+0 and 2) Additive identity = $0.x^{2}+0.x+0$

Finding Zeroes of a Polynomial

The zero of a polynomial is the value of the which polynomial gives zero. Thus, in order to find zeros of the polynomial, we simply equate polynomial to zero and find the possible values of variables.
Let P(x) be a given polynomial. To find zeros, set this polynomial equal to zero. i.e. P(x) = 0.Now, this becomes a polynomial equation. Solve this equation and find all the possible values of variables by factorizing the polynomial equation.
These are the values of x which make polynomial equal to zero; hence are called zeros of polynomial P(x). A number z is said to be a zero of a polynomial P(x) if and only if P(z) = 0.

Real and Complex Zeroes of Polynomials

When the roots of a polynomial are in the form of the real number, they are known as real zeros whereas complex numbers are written as a $\pm$ ib, where a is called real part and b is known as the imaginary part. The complex zeros are found in pairs, such as a + ib and a – ib.