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 \(p(-12)=0,\; p(-2)=6,\; p(4)=0,\; p(15)=0\), how many of the given x-values are zeroes (roots) of \(p\)?ExpertLevel 23If \(p(x)=x^2-hx\), what are the x-intercepts of its graph?ExpertLevel 23If a graph cuts the (x)-axis at ((-10,0)) and ((22,0)), what change is needed to make the zeroes equally distant from the (y)-axis?ExpertLevel 23If (p(x)=11(x+5)^2(x-14)), at how many distinct points will the graph meet the (x)-axis and which point will be a touching point?ExpertLevel 23If (p(x)=(x-4)(x-9)(x-15)), where will the graph be for (4<x<9)?ExpertLevel 23If (p(x)=x^2+gx+g^2) and (g\neq0), why will the graph not cut the (x)-axis?ExpertLevel 23If the vertex of a parabola is ((-14,0)) and it opens downward, how many distinct real zeroes are there?ExpertLevel 23If (p(x)=5x^2+20x+23), how will the graph be with respect to the (x)-axis?ExpertLevel 23If (p(x)=x^2-2dx+d^2-36), what will be the zeroes of the graph?ExpertLevel 23If a graph intersects the x-axis at (-4, 0), (6, 0) and (16, 0), what is the mean (average) of its zeros?ExpertLevel 23If the graph of a quadratic polynomial (p(x)) cuts the (x)-axis at (-2) and (5), how many zeroes does it have?ExpertLevel 24If the graph of a polynomial touches the x-axis at a point but does not cross it there, what does this indicate?ExpertLevel 24If the graph of a linear polynomial is parallel to the (x)-axis and not on the (x)-axis, how many zeroes will it have?ExpertLevel 24If p(x) = x^2 - 9, at which points does its graph cut the x-axis?ExpertLevel 24The graph of a quadratic polynomial touches the (x)-axis at only one point ((4,0)). What type of zeroes does it have?ExpertLevel 24If a parabola remains entirely above the x-axis and never touches it, how many real zeroes does it have?EasyLevel 24The graph shows p(2)=0. What is its geometric meaning?ExpertLevel 24If the graph of a cubic polynomial crosses the (x)-axis at (-1) and touches it at (2), how many distinct real zeroes are there?ExpertLevel 24What is the x-intercept of the graph of \(p(x)=3x+6\)?ExpertLevel 24If the x-intercepts of a polynomial graph are (-4,0), (1,0) and (6,0), what is the set of zeroes?ExpertLevel 24