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 \(p(-9)=0\), \(p(-4)=2\), \(p(2)=0\) and \(p(6)=0\), how many of the given x-values are zeros of \(p(x)\)?HardLevel 24If \(p(x)=x^2-cx\), what are the x-axis intersections (x-intercepts) of its graph?HardLevel 24If a graph cuts the (x)-axis at ((-6,0)) and ((14,0)), what change is needed to make the zeroes equally distant from the (y)-axis?HardLevel 24If (p(x)=7(x+3)^2(x-10)), at how many distinct points will the graph meet the (x)-axis and which point will be a touching point?HardLevel 24If (p(x)=(x-2)(x-5)(x-9)), where will the graph be for (2<x<5)?HardLevel 24If (p(x)=x^2+dx+d^2) and (d\neq0), why will the graph not cut the (x)-axis?HardLevel 24If the vertex of a parabola is ((-5,0)), what will be the distinct number of real zeroes?HardLevel 24If (p(x)=3x^2+12x+17), how will the graph be with respect to the (x)-axis?HardLevel 24If (p(x)=x^2-2bx+b^2-16), what will be the zeroes of the graph?HardLevel 24If a graph has x-axis intersections at (-2,0), (4,0) and (10,0), what is the mean of their zeroes?HardLevel 24If the axis of symmetry of a parabola is (x=-2) and one zero is (5), what will be the other zero?ExpertLevel 22If (p(x)=(x+5)^3(x-2)^2), where will the graph cross and where will it touch?ExpertLevel 22An upward-opening parabola has zeroes (-4) and (10). What will be the position of the graph at (x=0)?ExpertLevel 22If (p(x)=-(x-3)(x+7)(x-1)), on which side of the (x)-axis will the graph lie for (1<x<3)?ExpertLevel 22If (p(x)=x^2-2(a+3)x+(a+3)^2), at which (x)-value will the graph touch the (x)-axis?ExpertLevel 22If (p(x)=x^2-(2m-1)x+m(m-1)), what will be the (x)-axis intersections of the graph?ExpertLevel 22One zero of a parabola is −8, and the other zero is 12 more than the first. What is the axis of symmetry?MediumLevel 23If (p(-6)=0), (p(-2)<0), (p(5)=0), (p(9)>0), what is the distance between the given zeroes?ExpertLevel 22If (p(x)=x^3-7x^2+10x), what is the mean of the zeroes of the graph?ExpertLevel 22If (p(x)=(x-1)^2(x+4)^4), what are the number of distinct real zeroes and graph behavior?ExpertLevel 22