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 (p(x)=x^2+6x+18), what is the graphical meaning of (p(x)=0)?ExpertLevel 22If (p(x)=x^3-10x^2+24x), what is the set of zeroes of the graph?ExpertLevel 22If a graph cuts the (x)-axis at ((8,0)) and ((20,0)), what is the position of (x=14) between them?ExpertLevel 22If p(x) = 5x^2 - 50x + 125, how will the graph meet the x-axis?MediumLevel 22If (p(x)=-(x-9)(x+11)), on which side of the (x)-axis will the graph lie for (x<-11)?ExpertLevel 22If a graph touches the x-axis at (9, 0) and crosses it at (-5, 0), what is the product of the zeroes?ExpertLevel 22If the x-axis intersections of a graph are (0,0), (d,0), (−d,0), where \(d\neq0\), what is the product of the zeros?ExpertLevel 22If p(x) = x² − 17x + 72, what is the distance between the zeroes of the graph?MediumLevel 22If a downward-opening parabola has zeroes (-3) and (11), where will the graph be at (x=4)?ExpertLevel 22If (p(x)=x^{10}+1), what is the number of real zeroes of its graph?ExpertLevel 22If p(-10)=0, p(-1)=4, p(3)=0 and p(12)=0, how many of the given x-values are zeroes of p(x)?ExpertLevel 22If \(p(x)=x^2-ex\), what are the x-axis intercepts (x-intercepts) of its graph?ExpertLevel 22If a graph cuts the (x)-axis at ((-8,0)) and ((18,0)), what change is needed to make the zeroes equally distant from the (y)-axis?ExpertLevel 22If (p(x)=9(x+4)^2(x-12)), at how many distinct points will the graph meet the (x)-axis and which point will be a touching point?ExpertLevel 22If (p(x)=(x-3)(x-6)(x-11)), where will the graph be for (3<x<6)?ExpertLevel 22If (p(x)=x^2+fx+f^2) and (f\neq0), why will the graph not cut the (x)-axis?ExpertLevel 22If the vertex of a parabola is ((12,0)) and it opens upward, how many distinct real zeroes are there?ExpertLevel 22If (p(x)=4x^2+16x+21), how will the graph be with respect to the (x)-axis?ExpertLevel 22If (p(x)=x^2-2cx+c^2-25), what will be the zeroes of the graph?ExpertLevel 22If a graph intersects the x-axis at (-3,0), (5,0) and (13,0), what is the mean (average) of these zeroes?ExpertLevel 22