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)(x+6)), on which side of the (x)-axis will the graph lie for (x<-6)?HardLevel 22If a graph touches the x-axis at (4,0) and crosses it at (-2,0), what is the sum of the zeros?HardLevel 22If the graph of a polynomial intersects the x-axis at \((0,0)\) and \((a,0)\), where \(a \neq 0\), what is the product of its zeros?HardLevel 22If p(x)=x²−11x+30, what is the distance between the zeroes of the graph?MediumLevel 22If a parabola has zeroes (-1) and (7) and opens downward, where will the graph be outside the zeroes?HardLevel 22If (p(x)=x^4+1), what is the number of real zeroes of its graph?HardLevel 22If a polynomial has values \(p(-4)=0\), \(p(0)=3\), \(p(2)=0\), \(p(5)=0\), how many of the given x‑values are zeros (roots) of the polynomial?HardLevel 22If \(p(x)=x^2-ax\), what are the x-axis intercepts (x-intercepts) of its graph?HardLevel 22If a graph cuts the (x)-axis at ((-3,0)) and ((9,0)), what change is needed to make the zeroes equally distant from the (y)-axis?HardLevel 22If (p(x)=3(x-2)^2(x+1)), at how many distinct points will the graph meet the (x)-axis and which point will be a touching point?HardLevel 22The axis of symmetry of a parabola is (x=1) and one zero is (-5). What will be the other zero?HardLevel 23If (p(x)=(x-3)^2(x+4)), where will the graph touch and where will it cross?HardLevel 23If a parabola has zeroes (-8) and (2) and opens upward, where will the graph be at (x=-3)?HardLevel 23If (p(x)=-(x+2)(x-6)), on which side of the (x)-axis will the graph lie for (-2<x<6)?HardLevel 23If (p(x)=x^2-4kx+4k^2), at which (x)-value will the graph touch the (x)-axis?HardLevel 23If \(p(x)=x^2-(m+n)x+mn\), what are the x-axis intersections (x-intercepts) of its graph?HardLevel 23A parabola has one zero (9) and axis of symmetry (x=2). What is the other zero?HardLevel 23If p(-6)=0, p(-1)>0, p(4)=0 and p(7)<0, what is the product of the given zeroes?MediumLevel 23If p(x)=x^3-16x, at how many distinct points will the graph cut the x-axis?MediumLevel 23If (p(x)=(x+3)^2(x-5)^2), how will the graph behave with the (x)-axis?HardLevel 23