बहुपद के शून्यकों का ज्यामितीय अर्थ

A polynomial is an algebraic expression which is in the form of $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+......+a_{1}x^{1}+a_{0}$

where $a_{n},a_{n-1},a_{1},a_{0}$ are the real numbers, where $a_{n}\neq 0$. Also, we have learned the terms related to the polynomials, such as coefficients, terms, degree of a polynomial, zeroes of a polynomial and so on.

Geometrical Meaning of Zeroes of a Linear Polynomial

A linear polynomial is in the form $ax+b$, where $a\neq 0$. The graph of the linear equation, say $y=ax+b$ is a straight line. Assume that the graph $y=2x+3$ is a polynomial. It means that $y=2x+3$ is a straight line that passes through the points $(-2,-1)$ and $(2,7)$. Here are the the coordinates$(x,y)$ by taking a few values of $x$.

$x$	$-2$	$-1$	$0$	$1$	$2$
$y=2x+3$	$-1$	$1$	$3$	$5$	$7$

The graph of the linear equation $y=2x+3$ is given below:

From the graph, we can observe that graph $y=2x+3$ intersects the x-axis between $x=-1$ and $x=-2$. It means the straight line intersects the x-axis at the point $(-{\frac {3}{2}},0)$.

Hence $-{\frac {3}{2}}$ is the zero of the polynomial $y=2x+3$

In general, we can say that a linear polynomial $y=ax+b$, where $a\neq 0$, has exactly one zero. The zero of the linear polynomial is the x-coordinate of the point where the graph of $y=ax+b$ intersects at the x-axis.

Geometrical Meaning of Zeroes of Quadratic Polynomial:

We know that the standard form of a quadratic polynomial is ax²+bx+c, where a≠0. Now, let us understand the geometrical meaning of zeroes of quadratic polynomials with the help of an example.

Consider the quadratic equation $y=x^{2}-3x-4$

For the given quadratic equation, here are the the coordinates$(x,y)$ by taking a few values of $x$.

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$	$5$
$y=x^{2}-3x-4$	$6$	$0$	$-4$	$-6$	$-6$	$-4$	$0$	$6$

Hence, the coordinates formed are $(-2,6),(-1,0),(0,-4),(1,-6),(2,-6),(3,-4),(4,0)(5,6)$

Now, graph the points as shown below: