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
How does a polynomial graph meeting the x-axis differ in meaning from it meeting the y-axis?MediumLevel 23If the (x)-axis intersections of a graph are ((-8,0)) and ((3,0)), which is the correct pair of zeroes?MediumLevel 23A parabola cuts the x-axis at x = -4 and x = 6. What is the midpoint between these zeros?MediumLevel 23If \(p(x)=(x+7)(x-1)\), at which points will its graph intersect the x-axis?MediumLevel 23If the graph of a polynomial touches the x-axis at (2, 0) and crosses the x-axis at (5, 0), how many distinct real zeros does the polynomial have?MediumLevel 23In the table \(p(-5)=0,\ p(-1)=2,\ p(4)=0,\ p(6)=9\) are given. Which x-values are zeros of the polynomial \(p(x)\)?MediumLevel 23If p(0) = -3, which conclusion about the graph is correct?MediumLevel 23A quadratic graph does not cut the (x)-axis and its vertex is above the (x)-axis. What is the number of real zeroes?MediumLevel 23If \(p(x)=x^2-2x\), at which \(x\)-values does its graph intersect the \(x\)-axis?MediumLevel 23If the graph of a polynomial function cuts the x-axis at x = -2, x = 1 and x = 5, what is the sum of its zeros?MediumLevel 23If a graph passes through the points (0,4), (-3,0), (1,7) and (8,0), which points represent zeros of the polynomial?MediumLevel 23The graph of a linear polynomial is given by \(y=4x+12\). What is its zero (root)?MediumLevel 23If \(p(x)=(x-6)^2\), what will the graph do with the x-axis?MediumLevel 23If a polynomial graph cuts the (x)-axis at four distinct points, what can be its minimum possible degree?MediumLevel 23In which situation will x=0 NOT be a zero (root) of the polynomial?MediumLevel 23If \(p(-2)=0\) and \(p(3)=0\), which points must lie on the graph?MediumLevel 23If \(p(x)=x^2+4x+4\), how many real zeros are there?MediumLevel 23If the graph of a polynomial crosses the x-axis at (-7,0) and (2,0), what is the product of its zeros?MediumLevel 23If a graph has only one x-axis intersection at (-5, 0) and the graph crosses the axis there, what is the real zero?MediumLevel 23A graph intersects the x-axis at x = 2 and x = 10. What is the distance between these two zeros?MediumLevel 23