Write whether the following statement is 'True' or 'False'. Justify your answer.

Degree of zero polynomial is zero.

1. Some useful Identities:

(i) ${\left(a+b\right)}^2=a^2+b^2+2ab$

(ii) ${\left(a-b\right)}^2=a^2+b^2-2ab$

(iii) $\left(a+b\right)\left(a-b\right)=a^2-b^2$

(iv) ${\left(a+b+c\right)}^2=a^2+b^2+c^2+2ab+2bc+2ca$

(v) ${\left(a+b\right)}^3=a^3+b^3+3ab\left(a+b\right)$

(vi) ${\left(a-b\right)}^3=a^3-b^3-3ab\left(a-b\right)$

(vii) $a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$

(viii) $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$

(ix) $a^3+b^3+c^3-3abc=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)$

(x) $a^3+b^3+c^3-3abc=\frac{1}{2}\left(a+b+c\right)\left\{{\left(a-b\right)}^2+{\left(b-c\right)}^2+\left(c-a\right){}^2\right\}$

(xi) If a+b+c=0 then a3+b3+c3=3abc

2. The general form of a Polynomial:

An algebraic expression of the form $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots \dots +a_{1}x+a_0$ where $a_0, a_1, a_2, \dots , a_n$ are constants, is known as a polynomial in variable $x$.

3. Terms of a Polynomial:

In the polynomial $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots \dots +a_{1}x+a_0$ each of $a_n{x}^n, a_{n-1}{x}^{n-1}, \dots , a_{1}x, a_0$ is called its term and $a_n{x}^n, a_n\ne 0$ called the leading term. a0 is known as the constant term.

4. Degree of a Polynomial:

A polynomial $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots \dots +a_{1}x+a_0$ is of degree $n,$ if an≠05. .

5. Classification of Polynomials based on Degree:

(i) A polynomial of degree $1$ is called a linear polynomial. For example, $f\left(x\right)=ax+b, a\ne 0$ is a linear polynomial.

(ii) A polynomial of degree $2$ is called a quadratic polynomial. Thus, $f\left(x\right)=a{x}^2+bx+c, a\ne \mathrm{ }0$; is the general form of a quadratic polynomial.

(iii) A polynomial of degree $3$ is called a cubic polynomial. Thus, $f\left(x\right)=a{x}^3+b{x}^2+cx+d, \mathrm{a}\ne \mathrm{ }0$; is the general form of a cubic polynomial.

6. Zeros of a Polynomial:

(i) A real number α is a zero (or root) of a polynomial $f\left(x\right)$, if $f\left(\alpha \right)=0$.

(ii) A polynomial of degree $n$ has maximum of $n$ roots.

(iii) A linear polynomial $f\left(x\right)=ax+b, a\ne 0$ has a unique root given by x=-ba

(iv) A non-zero constant polynomial has no root.

(v) Every real number is a root of the zero polynomial.

(vi) If $f\left(x\right)$ is a polynomial with integral coefficients and the leading coefficient is $1,$ then any integer root of $f\left(x\right)$ is a factor of the constant term.

(vii) Let $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots \dots +a_{1}x+a_n, a_n\ne 0$ be a polynomial. Then, bc (a rational fraction in lowest terms) is a root of $f\left(x\right),$ if $b$ is a factor of constant term a0 and $c$ is a factor of the leading term an.

7. Remainder Theorem:

Let $f\left(x\right)$ be a polynomial of degree greater than or equal to one and $a$ be any real number. If $f\left(x\right)$ is divisible by $(x-a),$ then the remainder is equal to $f\left(a\right).$

8. Factor Theorem:

Let $f\left(x\right)$ be a polynomial of degree greater than or equal to one and $a$ be real number such that $f\left(a\right)=0,$ then $(x-a)$ is a factor of $f\left(x\right).$ Conversely, if $(x-a)$ is a factor of $f\left(x\right),$ then $f\left(a\right)=0.$