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
The axis of symmetry of a parabola is (x=4) and one zero is (-2). What will be the other zero?HardLevel 24If (p(x)=(x+6)^2(x-2)), where will the graph touch and where will it cross?HardLevel 24If a parabola has zeroes (-7) and (5) and opens upward, where will the graph be at (x=-1)?HardLevel 24If (p(x)=-(x+5)(x-1)), on which side of the (x)-axis will the graph lie for (-5<x<1)?HardLevel 24If \(p(x)=x^2-6kx+9k^2\), at which x-value will its graph touch the x-axis?HardLevel 24If (p(x)=x^2-(u+v)x+uv), what will be the (x)-axis intersections of the graph?HardLevel 24A parabola has one zero (11) and axis of symmetry (x=3). What will be the other zero?HardLevel 24If (p(-8)=0), (p(-2)>0), (p(3)=0) and (p(9)<0), what is the sum of the given zeroes?HardLevel 24If (p(x)=x^3-25x), at how many distinct points will the graph cut the (x)-axis?HardLevel 24If (p(x)=(x+1)^2(x-8)^2), how will the graph behave with the (x)-axis?HardLevel 24If p(x) = (x - 4)(x + 7)^3, what are the distinct zeroes?MediumLevel 24If the (x)-axis intersections of a parabola are ((-11,0)) and ((5,0)), what is the coordinate of the midpoint?HardLevel 24If a graph cuts the x-axis at x = -12, x = -3 and x = 8, what is the range of the zeroes?EasyLevel 24If (p(x)=x^2-8x-33), what are the (x)-axis intersections of the graph?HardLevel 24If the graph of a polynomial function y = p(x) passes through the point (0,0) on the y-axis, which conclusion about its zeroes is correct?HardLevel 24If (p(x)=x^2+14x+49), at which point will the graph touch the (x)-axis?HardLevel 24If the (x)-axis intersections of a graph are ((m,0)), ((n,0)), ((r,0)), what will be the mean of the zeroes?HardLevel 24If (p(x)=x^2-8x+20), why will the graph not cut the (x)-axis?HardLevel 24If (p(x)=x(x+4)^2(x-6)), what will be the graph behavior at (x=-4)?HardLevel 24If the zeroes of a parabola are (c-7) and (c+3), what will be its axis of symmetry?HardLevel 24