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
A graph cuts the (x)-axis three times. What can be the minimum possible degree of that polynomial?ExpertLevel 24Which statement is correct about the graph of the quadratic polynomial (p(x)=x^2+4)?ExpertLevel 24If the graph of a polynomial crosses the x-axis at the origin (0,0), which conclusion is certain?ExpertLevel 24If the graph of a polynomial meets the x-axis at the point (7, 0), what is the value of p(x) at that point?ExpertLevel 24If a quadratic graph cuts the (x)-axis at two distinct points, what kind of real zeroes does it have?ExpertLevel 24If p(x) = 2(x-1)(x+4), what are the x-intercepts of its graph?ExpertLevel 24In which situation is \\(x=3)\\ called a zero of the polynomial \\(p(x))\\?ExpertLevel 24A polynomial's graph crosses the \(x\)-axis at \\((-5,0)\\) and touches the \(x\)-axis at \\((2,0)\\). Based on this information, what is the number of real (distinct) zeros?ExpertLevel 24If (p(x)=(x+2)^2), how will its graph behave with the (x)-axis?ExpertLevel 24If the vertex of a quadratic polynomial graph lies on the x-axis, what is correct about its real zeroes?EasyLevel 24If the graph of a polynomial cuts the \(x\)-axis at \(a\) and \(b\) with \(a \neq b\), what are the values of \(p(a)\) and \(p(b)\)?ExpertLevel 24The graph of a polynomial intersects the y-axis at the point (0, −8). Which conclusion about the polynomial's zeros is correct?ExpertLevel 24If \(p(x)=-(x-3)(x+1)\), what are the zeros of its graph?ExpertLevel 24If a parabola opens downward and cuts the (x)-axis at two points, how many real zeroes will it have?ExpertLevel 24The graph cuts the x-axis at (−2, 0), (0, 0), and (5, 0). Which statement is correct?EasyLevel 24If (p(x)=x^2-6x+9), how will the graph meet the (x)-axis?ExpertLevel 24If the graph of a polynomial cuts the (x)-axis at four distinct points, which statement about its degree is correct?ExpertLevel 24How many real zeros does the constant polynomial \(p(x)=5\) have?ExpertLevel 24What is the graph of the zero polynomial \(p(x)=0\)?ExpertLevel 24If the graph of a polynomial crosses the x-axis only at x = -6 and does not meet the axis anywhere else, what is the polynomial's zero (root)?ExpertLevel 24