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
The graph of a quadratic polynomial opens downward and cuts the (x)-axis at two points. What is the number of real zeroes?MediumLevel 22If \(p(-3)=0\), \(p(0)=5\) and \(p(4)=0\), at which points will the graph intersect the x-axis?MediumLevel 22A polynomial graph cuts the (x)-axis twice and the (y)-axis once. How many real zeroes does it have?MediumLevel 22If a parabola does not cut the (x)-axis but cuts the (y)-axis at ((0,-2)), what is the number of real zeroes?MediumLevel 22The table gives \(p(-2)=6,\; p(1)=0,\; p(3)=-4,\; p(5)=0\). Which x-values are zeroes of the polynomial \(p(x)\)?MediumLevel 22If the graph of a polynomial crosses the x-axis at \(x=-2\) and only touches it at \(x=3\), what are the distinct real zeros?MediumLevel 22Which of the following graphs will have exactly one real zero?MediumLevel 22If \(p(x)=x^2-16\), what are the x-axis intersections (x-intercepts) of its graph?MediumLevel 22If \(p(x)=x^2+9\), how many times will its graph cut the x-axis?MediumLevel 22If (p(x)=(x+1)^2), how will the graph meet the (x)-axis?MediumLevel 22A parabola cuts the (x)-axis at ((-5,0)) and ((1,0)). What is the greater zero?MediumLevel 22If the (x)-axis intersections of a graph are ((m,0)) and ((n,0)), what is the product of the zeroes?MediumLevel 22The points ((2,0)), ((3,4)), ((4,0)) and ((5,-1)) lie on a polynomial graph. Why will (3) and (5) not be called zeroes?MediumLevel 22If a polynomial graph appears to cut the (x)-axis four times, for which type of polynomial can this be possible?MediumLevel 22In which situation will a linear polynomial have no real zero?MediumLevel 22If p(2) is positive and p(5) is negative, which statement about the graph is most appropriate?MediumLevel 22If \(p(-1)<0\), \(p(0)<0\) and \(p(2)=0\), which of the given numbers is definitely a zero of the polynomial?MediumLevel 22If the graph of a quadratic polynomial cuts the x-axis at (-2, 0) and (7, 0), what are the values of p(-2) and p(7)?MediumLevel 22A graph cuts the x-axis at x = -3 and x = 9. What is the distance between these zeroes?MediumLevel 22If the zeroes are (-2) and (6), what is the midpoint of the (x)-axis intersection points?MediumLevel 22