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 ax2 + bx +c, has nil, one or two zeroes

E.g. There are no Zeroes of Quadratic polynomial p(x) = x2 + 1 & thus the graph of this polynomial doesn’t intersect x axis.

E.g. There is one Zero of Quadratic polynomial p(x) = x2 -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) = x2 – 4, that is +2, -2 & thus the graph of this polynomial intersect x axis at two place, x=2 & x=-2.

Cubic polynomial ax3 + bx2 +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) = x3, 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) = x3 – x2, 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) = x3 -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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