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 a graph cuts the (x)-axis at ((-2,0)) and ((10,0)), what is the midpoint of these intersection points?MediumLevel 24The (x)-axis intersections of a parabola are ((1,0)) and ((7,0)). Through which (x)-value can its axis of symmetry pass?MediumLevel 24If (p(x)=2(x-4)(x+1)), how does the outside multiplier (2) affect the (x)-axis intersections of the graph?MediumLevel 24If \(p(x)=x^3-4x\), at which x-values does its graph meet the x-axis?MediumLevel 24In which situation will 0 be a real zero of a polynomial?MediumLevel 24If a polynomial graph cuts the (x)-axis at five distinct points, what can be the minimum degree of the polynomial?MediumLevel 24If (p(x)=-2(x+5)^2), where will the graph meet the (x)-axis?MediumLevel 24If (a, 0) and (b, 0) are intersection points of a graph and a < b, what is the smaller zero?MediumLevel 24A table gives p(-4) = −2, p(-1) = 0, p(3) = 0, p(6) = 5. What is the product of the polynomial's zeroes from these values?MediumLevel 24If a graph intersects the x-axis at the point \((r,0)\) with \(r>0\), which statement about the zero is correct?MediumLevel 24If p(x) = x² − 3x − 10, what are the x-axis intersections of the graph?MediumLevel 24Which graph will give exactly two distinct real zeroes?MediumLevel 24If the points ((3,0)) and ((3,0)) show the same touch twice on a graph, how many distinct zeroes are there?MediumLevel 24If (p(x)=x(x+6)), what are the zeroes of the graph?MediumLevel 24If a parabola stays below the (x)-axis and never touches it, how many real solutions does (p(x)=0) have?MediumLevel 24If a graph cuts the x-axis at the points (-3, 0), (0, 0) and (6, 0), what is the sum of its zeroes?MediumLevel 24If \(p(x)=x^2-49\), at which points does its graph intersect the x-axis?MediumLevel 24If the real zeroes of a polynomial are (5), (5) and (−1), what are the distinct real zeroes?MediumLevel 24If the vertex of a parabola is ((-2,0)) and it opens downward, what is correct about the zero?MediumLevel 24If a graph intersects the x-axis at (-12, 0) and (-4, 0), which zero is greater?MediumLevel 24