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+8x+16), at which point will the graph touch the (x)-axis?HardLevel 22If the (x)-axis intersections of a polynomial graph are ((r,0)), ((s,0)), ((t,0)), what will be the sum of the zeroes?HardLevel 22If (p(x)=x^2-6x+13), why will the graph not cut the (x)-axis?HardLevel 22If the graph of a cubic polynomial touches or crosses the (x)-axis at two distinct points, how many distinct real zeroes will it have?HardLevel 22If (p(x)=x(x-2)^2(x+3)), what will be the graph behavior at (x=2)?HardLevel 22If the zeroes of a parabola are (a-2) and (a+4), what will be its axis of symmetry?HardLevel 22If a graph has zeros −2, 3 and 7, which is the correct set of x-axis intersection points?HardLevel 22If (p(x)=(x+2)(x-1)(x-6)), what will be the sign of (p(x)) for (1<x<6)?HardLevel 22If (p(x)=x^2-5x+k) has one zero (2), what will be the other zero and the (x)-axis intersections?HardLevel 22If a graph cuts the (x)-axis at (x=-5) and (x=5), which is the strongest conclusion about the zeroes?HardLevel 22If \(p(x)=4x^2-25\), what are the x-axis intercepts of its graph?HardLevel 22If the points meeting the x-axis on a polynomial graph are written as (1,0), (1,0), (4,0), how many distinct real zeroes are there?MediumLevel 22If the graph of (p(x)=x^2+px+q) cuts the (x)-axis at ((-2,0)) and ((5,0)), what will be the values of (p) and (q)?HardLevel 22If a parabola cuts the (x)-axis at two points and cuts the (y)-axis at ((0,12)), what is the number of real zeroes?HardLevel 22If (p(x)=x^4-1), what are the real (x)-axis intersections of the graph?HardLevel 22If a polynomial graph cuts the (x)-axis at three distinct points, what can be the minimum possible degree of the polynomial?HardLevel 22If (p(x)=x^2-2x+5), what will be the graphical meaning of (p(x)=0)?HardLevel 22If (p(x)=x^3-4x^2-5x), what is the set of zeroes of the graph?HardLevel 22If a graph cuts the x-axis at (2,0) and (8,0), what is the position of x=5 between them?MediumLevel 22If p(x)=2x^2+12x+18, how will the graph meet the x-axis?MediumLevel 22