Mathematics
Geometrical meaning of the zeroes of a polynomial.
बहुपद के शून्यकों का ज्यामितीय अर्थ
In this Class 10 Mathematics topic from Polynomials, students understand the geometrical meaning of a polynomial’s zeroes by connecting algebraic expressions with their graphs. A zero is represented by the x-coordinate where the graph meets or touches the x-axis. Students learn how the number of points of intersection indicates the number of real zeroes, and interpret graphs of linear, quadratic, and other polynomial functions to relate their shapes and x-intercepts to solutions of p(x) = 0.
Practice questions
If the zeroes of a graph are (-12), (5), (14), which point does not show a zero on the graph?ExpertLevel 23If (p(x)=(x+9)(x-5)(x-12)), what will be the sign of (p(x)) for (5<x<12)?ExpertLevel 23If (p(x)=x^2-13x+k) has one zero (6), what will be the other zero and the (x)-axis intersections?ExpertLevel 23If a graph cuts the (x)-axis at (x=-12) and (x=12), what are the product and sum of the zeroes?ExpertLevel 23If \(p(x)=36x^2-49\), what are the x-axis intersections of its graph?ExpertLevel 23If the points meeting the (x)-axis are written as ((-11,0)), ((-11,0)), ((4,0)), ((4,0)), how many distinct real zeroes are there?ExpertLevel 23If the graph of p(x) = x² + px + q cuts the x-axis at (−8, 0) and (5, 0), what are p and q?MediumLevel 23A parabola cuts the (x)-axis at two points and cuts the (y)-axis at ((0,25)). What is the number of real zeroes?ExpertLevel 23If (p(x)=x^4-1296), what are the real (x)-axis intersections?ExpertLevel 23If a polynomial cuts the (x)-axis at eight distinct points, what is the minimum possible degree?ExpertLevel 23If (p(x)=x^2+16x+80), what is the graphical meaning of (p(x)=0)?ExpertLevel 23If (p(x)=x^3-12x^2+35x), what is the set of zeroes of the graph?ExpertLevel 23If a graph cuts the (x)-axis at ((10,0)) and ((26,0)), what is the position of (x=18) between them?ExpertLevel 23If p(x) = 6x² − 72x + 216, how will the graph meet the x-axis?MediumLevel 23If (p(x)=-(x-11)(x+13)), on which side of the (x)-axis will the graph lie for (x<-13)?ExpertLevel 23If a graph touches the x-axis at (11, 0) and crosses it at (-6, 0), what is the product of the zeroes of the polynomial?ExpertLevel 23If the x-axis intersections of a graph are (0, 0), (g, 0), and (−g, 0), where g ≠ 0, what is the sum of the zeroes?EasyLevel 23If p(x) = x² − 19x + 90, what is the distance between the zeroes of the graph?MediumLevel 23If a downward-opening parabola has zeroes (-5) and (13), where will the graph be at (x=2)?ExpertLevel 23If (p(x)=x^{12}+1), what is the number of real zeroes of its graph?ExpertLevel 23