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+a)^3(x-b)^2), where (a\neq -b), what are the distinct zeroes?ExpertLevel 22The (x)-axis intersections of a parabola are ((-13,0)) and ((7,0)). What is the coordinate of the midpoint of its zeroes?ExpertLevel 22If a graph cuts the (x)-axis at (x=-15), (x=-5), (x=4), and (x=11), what is the range of the zeroes?ExpertLevel 22If (p(x)=x^2-9x-52), what are the (x)-axis intersections of the graph?ExpertLevel 22If a graph cuts the (y)-axis at the origin and cuts the (x)-axis at (x=6), what are the zeroes?ExpertLevel 22If (p(x)=x^2-18x+81), at which point will the graph touch the (x)-axis?ExpertLevel 22If the x-axis intersections of a graph are \\((r-1,0),\ (r+2,0),\ (r+5,0)\\), what is the mean of the zeroes?ExpertLevel 22If (p(x)=x^2-10x+29), why will the graph not cut the (x)-axis?ExpertLevel 22If (p(x)=x(x-7)^4(x+2)^3), what will be the graph behavior at (x=7)?ExpertLevel 22If the zeroes of a parabola are (t-9) and (t+5), what will be its axis of symmetry?ExpertLevel 22If the zeroes of a graph are (-8), (3), (12), which point does not show a zero on the graph?ExpertLevel 22If (p(x)=(x+6)(x-4)(x-10)), what will be the sign of (p(x)) for (4<x<10)?ExpertLevel 22If (p(x)=x^2-11x+k) has one zero (4), what will be the other zero and the (x)-axis intersections?ExpertLevel 22If a graph cuts the (x)-axis at (x=-10) and (x=10), what are the product and sum of the zeroes?ExpertLevel 22If \(p(x)=25x^2-36\), what are the x-axis intersections of its graph?ExpertLevel 22If the points meeting the (x)-axis are written as ((-7,0)), ((-7,0)), ((2,0)), ((2,0)), how many distinct real zeroes are there?ExpertLevel 22If the graph of p(x) = x² + px + q cuts the x-axis at (−9, 0) and (4, 0), what will p and q be?HardLevel 23A parabola cuts the (x)-axis at two points and cuts the (y)-axis at ((0,-20)). What is the number of real zeroes?ExpertLevel 22If (p(x)=x^4-625), what are the real (x)-axis intersections?ExpertLevel 22If a polynomial cuts the (x)-axis at seven distinct points, what is the minimum possible degree?ExpertLevel 22