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=2) and one zero is (-3). What will be the other zero?HardLevel 22If a polynomial graph crosses at (x=-2), touches at (x=1), and crosses at (x=5), how many distinct real zeroes are there?HardLevel 22If (p(x)=(x+4)^2(x-3)), where will the graph touch and where will it cross the (x)-axis?HardLevel 22The zeroes of a parabola are (-6) and (2). If it opens upward, where will the graph be relative to the (x)-axis at (x=-2)?HardLevel 22If (p(x)=-(x-1)(x-5)), on which side of the (x)-axis will the graph lie for (1<x<5)?HardLevel 22If a polynomial graph cuts the x-axis at (−4, 0), (2, 0), and (9, 0), what is the product of these zeroes?HardLevel 22If a quadratic graph has vertex ((3,-4)) and opens upward, what will be the number of real zeroes?HardLevel 22If a quadratic graph has vertex ((-1,0)) and opens downward, how will it meet the (x)-axis?HardLevel 22If \(p(x)=x^2-2kx+k^2\), at which x-value will its graph touch the x-axis?HardLevel 22If (p(x)=x^2-(a+b)x+ab), what will be the (x)-axis intersections of the graph?HardLevel 22A parabola has one zero (4) and axis of symmetry (x=-1). What will be the other zero?HardLevel 22If on a graph (p(-5)>0), (p(-2)=0), (p(1)<0), (p(6)=0), what is the sum of the zeroes among the given values?HardLevel 22If the graph of a polynomial crosses the x-axis at \\(x=-3\\) and only touches it at \\(x=2\\), which statement is correct?HardLevel 22If (p(x)=x^3-9x), at how many distinct points can the graph cut the (x)-axis?HardLevel 22If (p(x)=(x-2)^2(x+5)^2), how will the graph behave with the (x)-axis?HardLevel 22If (p(x)=(x+1)(x-4)^3), what are the distinct zeroes?HardLevel 22If the (x)-axis intersections of a parabola are ((-7,0)) and ((1,0)), what is the coordinate of the midpoint of its zeroes?HardLevel 22If a graph cuts the (x)-axis at (x=-8), (x=-2), (x=4), what is the range of the zeroes?HardLevel 22If (p(x)=x^2+2x-15), what are the (x)-axis intersections of the graph?HardLevel 22A student says that if the graph of a polynomial meets the y-axis at (0,0) then x=0 cannot be a zero (root) because that point lies on the y-axis. What is the correct conclusion?HardLevel 22