An equation of the form \[f(x)=a_0x^n+a_1x^{n-1}+…+a_n=0\;\;\;(a_0≠0)\] where $a_0,a_1,…,a_n$ are constants and $n$ is a non negative integer is called a polynomial equation of degree $n$.
Polynomial of Degree n in x
Let a function $f$ be defined by \[f(x)=a_0x^n+a_1x^{n-1}+…+a_n\;\;\;(a_0≠0)\] Here, $n$ is a non negative integer and $a_0,a_1,…,a_n$ are all constants. This type of function is called a rational integral function or polynomial of degree $n$ in $x$.
The constants $a_0,a_1,…,a_n$ are called the coefficients of $x^0,x^{n-1},…,x^0$ respectively, and each of $a_0x^n,a_1x^{n-1},…,a_n$ is called a term of the polynomial. The term $a_0x^n$ is called the leading term of the polynomial.
A polynomial is also known as a quantic. The quantics of various successive degrees have special names. The polynomials of degrees one, two, three and four are called linear (first degree), quadratic (or quadric), cubic and biquardic (or quartic) respectively.
Thus, the polynomials defined by \[\begin{array}{l} & g(x) &=ax+b, \\ & G(x) &= ax^2+bx+c, \\ & h(x) &= ax^3+bx^2+cx+d, \\ \text{and}, & H(x) &= ax^4+bx^3+cx^2+dx+e \end{array}\] are linear, quadratic, cubic and biquadratic respectively.
Consider a linear function, \[g(x)=2x-4\]
Suppose for certain value of $x=a$, $g(a)=0$. Then, the value $x=a$ is called a zero of the polynomial. In the function $g(x)$, the value $x=2$ is a zero of the linear function defined by $g(x)=2x-4$, because \[g(2)=2×2-4=0\]
Also, it may be noted that there is no other zero i.e. there is no other value of $x$ which makes $g(x)=0$.
Similarly, consider a quadratic function, \[G(x)=x^2+2x-3\]
In this quadratic function, the zeros are $x=1$ and $x=-3$. Here, the number of zeros is exactly two. Hence, a polynomial may have one or more zeros depending upon the degree of the polynomial.
Polynomial Equation
A polynomial of degree $n$ in $x$ is defined by \[f(x)=a_0x^n+a_1x^{n-1}+…+a_{n_1}x+a_n\;\;\;(a_0≠0)\] where, $n$ is a non negative integer and $a_0,a_1,…,a_{n-1},a_n$ are all constants.
Suppose for some values of $x$, $f(x)=0$. Then, \[f(x)=0\] is called a general equation of degree $n$ in $x$. The values of $x$ for which $f(x)=0$ are known as the solutions of the equation. This equation is also known as a polynomial equation of degree $n$ in $x$.
Thus, some polynomial equations of degree one, two, three and four are respectively defined as;
Linear equation: \[ax+b=0\] Quadratic equation: \[ax^2+bx+c=0\] Cubic Equation: \[ax^3+bx^2+cx+d=0\] Biquadratic equation: \[ax^4+bx^3+cx^2+dx+e=0\]
Regarding polynomial equations, the fundamental theorem of algebra is ‘Every equation has at least one root‘.
Theorem: Every equation of degree n in x has n roots, and no more.
A polynomial equation of degree $n$ in $x$ is defined by \[f(x)=a_0x^n+a_1x^{n-1}+…+a_n=0\;\;\;(a_0≠0)\] where, $n$ is a non negative integer and $a_0,a_1,…,a_n$ are all constants.
From the Fundamental Theorem of Algebra, the equation $f(x)=0$ has a real or imaginary root. Let this root be $\alpha_1$. Then, by Factor Theorem, $x-\alpha_1$ is a factor of $f(x)$.
\[\therefore f(x)=(x-\alpha_1)\phi_1(x)\] where, $\phi_1(x)=a_0x^{n-1}+a_1x^{n-2}+…$ is a polynomial of degree $n-1$.
Again, the equation $\phi_1(x)=0$ has a real or imaginary root. Let this root be $\alpha_2$. Then, $x-\alpha_2$ is a factor of $\phi_1(x)$. \[\therefore\phi_1(x)=(x-\alpha_2)\phi_2(x)\] where, $\phi_2(x)=a_0x^{n-2}+a_1x^{n-3}+…$ is a polynomial of degree $n-2$. Thus, \[f(x)=(x-\alpha_1)(x-\alpha_2)\phi_2(x)\]
Proceeding in this way, we obtain \[f(x)=a_0(x-\alpha_1)(x-\alpha_2)(x-\alpha_3)…(x-\alpha_n)\]
Hence, $f(x)=0$ has $n$ solutions (or roots), since $f(x)$ becomes zero when $x$ has any one of the values \[\alpha_1,\alpha_2,\alpha_3,…,\alpha_n.\]
Also, the equation cannot have more than one root; for if $x$ has any values different from any of the quantities $\alpha_1,\alpha_2,\alpha_3,…,\alpha_n$, all the factors on the right are different from zero. Hence for that value of $x$, $f(x)≠0$.
This theorem may be used to deduce that;
- A quadratic equation has two and only two roots.
- A cubic equation has three and only three roots.
- A biquadratic equation has four and only four roots.
It may also be used in the investigation of the relations between the roots and the coefficients of any equation.
Next: Quadratic Equation