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+10x+29), what is the graphical meaning of (p(x)=0)?HardLevel 23If (p(x)=x^3-6x^2+5x), what is the set of zeroes of the graph?HardLevel 23If a graph cuts the (x)-axis at ((4,0)) and ((12,0)), what is the position of (x=8) between them?HardLevel 23If p(x) = 3x^2 - 18x + 27, how will the graph meet the x-axis?MediumLevel 23If (p(x)=-(x-4)(x+8)), on which side of the (x)-axis will the graph lie for (x<-8)?HardLevel 23If a graph touches the x-axis at \\((5,0)\\) and crosses it at \\((-3,0)\\), what is the sum of the zeroes?HardLevel 23If the (x)-axis intersections of a graph are ((0,0)) and ((b,0)), where (b\neq0), what will be the sum of the zeroes?HardLevel 23If p(x) = x² − 13x + 42, what is the distance between the zeroes of the graph?MediumLevel 23If a parabola has zeroes (-2) and (8) and opens downward, where will the graph be outside the zeroes?HardLevel 23If (p(x)=x^6+1), what is the number of real zeroes of its graph?HardLevel 23If \(p(-7)=0\), \(p(-3)=2\), \(p(1)=0\) and \(p(4)=0\), how many of the given x‑values are zeroes of \(p(x)\)?HardLevel 23If \(p(x)=x^2-bx\), at which points does its graph intersect the x-axis?HardLevel 23If a graph cuts the (x)-axis at ((-4,0)) and ((10,0)), what change is needed to make the zeroes equally distant from the (y)-axis?HardLevel 23If (p(x)=5(x+2)^2(x-7)), at how many distinct points will the graph meet the (x)-axis and which point will be a touching point?HardLevel 23If (p(x)=(x-1)(x-3)(x-5)), where will the graph be for (1<x<3)?HardLevel 23If (p(x)=x^2+cx+c^2) and (c\neq0), why will the graph not cut the (x)-axis?HardLevel 23If the vertex of a parabola is ((4,0)), what will be the distinct number of real zeroes?HardLevel 23If (p(x)=2x^2+8x+11), how will the graph be with respect to the (x)-axis?HardLevel 23If (p(x)=x^2-2ax+a^2-9), what will be the zeroes of the graph?HardLevel 23If the x-axis intersections of a graph are (-1,0), (3,0), (7,0), what is the mean of their zeroes?HardLevel 23