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 the axis of symmetry of a parabola is (x=5) and one zero is (-1), what will be the other zero?ExpertLevel 23If (p(x)=(x-4)^4(x+3)^3), where will the graph touch and where will it cross?ExpertLevel 23An upward-opening parabola has zeroes (-9) and (3). What will be the position of the graph at (x=-4)?ExpertLevel 23If (p(x)=-(x+8)(x-2)(x-6)), on which side of the (x)-axis will the graph lie for (2<x<6)?ExpertLevel 23If (p(x)=x^2-2(b-4)x+(b-4)^2), at which (x)-value will the graph touch the (x)-axis?ExpertLevel 23If (p(x)=x^2-(2n+3)x+n(n+3)), what will be the (x)-axis intersections of the graph?ExpertLevel 23One zero of a parabola is −10, and the other zero is 18 more than the first. What is the axis of symmetry?MediumLevel 23If (p(-7)=0), (p(-3)<0), (p(4)=0), (p(8)>0), what is the distance between the given zeroes?ExpertLevel 23If p(x) = x³ − 9x² + 18x, what is the mean of the zeroes of the graph?MediumLevel 23If (p(x)=(x+2)^6(x-5)^2), what are the number of distinct real zeroes and graph behavior?ExpertLevel 23If (p(x)=(x-c)^5(x+d)^2), where (c\neq -d), what are the distinct zeroes?ExpertLevel 23The (x)-axis intersections of a parabola are ((-17,0)) and ((9,0)). What is the coordinate of the midpoint of its zeroes?ExpertLevel 23If a graph cuts the (x)-axis at (x=-18), (x=-6), (x=5), and (x=13), what is the range of the zeroes?ExpertLevel 23If (p(x)=x^2-13x-68), what are the (x)-axis intersections of the graph?ExpertLevel 23If a graph cuts the (y)-axis at the origin and cuts the (x)-axis at (x=-9), what are the zeroes?ExpertLevel 23If (p(x)=x^2+20x+100), at which point will the graph touch the (x)-axis?ExpertLevel 23If the (x)-axis intersections of a graph are ((s-4,0)), ((s+1,0)), ((s+7,0)), what is the mean of the zeroes?ExpertLevel 23If (p(x)=x^2-14x+53), why will the graph not cut the (x)-axis?ExpertLevel 23If (p(x)=x(x+6)^4(x-3)^5), what will be the graph behavior at (x=3)?ExpertLevel 23If the zeroes of a parabola are (q-11) and (q+7), what will be its axis of symmetry?ExpertLevel 23