Class 10 Maths Polynomials | Geometrical Meaning of Zeroes of Polynomials |

**Geometrical Meaning of Zeroes of Polynomials**

A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.

Polynomials can easily be represented graphically.

Zero of polynomial p(x) is x-coordinate of point where graph of p(x) intersects x-axis. Polynomial p(x) intersects the x-axis @ x=2, thus zero of this polynomial is 2.

Linear polynomial ax + b, a ≠ 0, has exactly one zero

E.g. Zero of linear polynomial p(x) = 2x -6 is 3 & thus the graph of this polynomial intersect x axis only once.

Quadratic polynomial ax^{2} + bx +c, has **nil**, **one** or **two** zeroes

E.g. There are no Zeroes of Quadratic polynomial p(x) = x^{2} + 1 & thus the graph of this polynomial doesn’t intersect x axis.

E.g. There is one Zero of Quadratic polynomial p(x) = x^{2} -4x+4, that is 2 & thus the graph of this polynomial intersect x axis once at x=2.

E.g. There are Two Zeros of Quadratic polynomial p(x) = x^{2} – 4, that is +2, -2 & thus the graph of this polynomial intersect x axis at two place, x=2 & x=-2.

Cubic polynomial ax^{3} + bx^{2} +cx +d, has **one**, **two** or **three** zeroes. There can’t be any cubic polynomial with **Nil** zeroes.

E.g. There is one Zero of Cubic polynomial p(x) = x^{3}, that is 0 & thus the graph of this polynomial intersect x axis once at x=0.

E.g. There are Two Zeros of Cubic polynomial p(x) = x^{3} – x^{2}, that is 0 & 1, and thus the graph of this polynomial intersect x axis at two place, x=0 & x=1

E.g. There are three Zeroes of Cubic polynomial p(x) = x^{3} -4x, that is 0,+2 & -2, and thus the graph of this polynomial intersect x axis at three place, x=0, x=+2 & x=-2.

In general, given a polynomial p(x) of degree n, the graph of y = p(x) intersects the x- axis at at most n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.

Also, Any polynomial of odd degree will never have nil zeroes

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