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 \(p(x)=3(x+4)(x-2)\), what effect does the constant factor 3 have on the zeros of \(p(x)\)?MediumLevel 23If the vertex of a parabola is ((1,0)) and it opens upward, what will be its relation with the (x)-axis?MediumLevel 23If the polynomial \(p(x)\) is continuous and \(p(-1)>0\) while \(p(4)<0\), which of the following statements is true?MediumLevel 23Which statement is correct when the graph of a polynomial touches the x-axis at the point (a,0)?MediumLevel 23If p(x) = x² − 6x + 9, at which point will its graph touch the x-axis?MediumLevel 23If a polynomial graph crosses the x-axis three times and touches it once, what is the number of distinct real zeroes?MediumLevel 23If the same touching point ((4,0)) is written twice on a graph, how many distinct zeroes are there?MediumLevel 23If \(p(x)=x^2-1\), which point on the graph does not represent a zero (root) of the polynomial?MediumLevel 23The zeroes of a polynomial are -9 and -2. Which zero is greater?MediumLevel 23If a graph intersects the x-axis at \((-10,0)\) and \((0,0)\), what is the product of its zeroes?MediumLevel 23If \(p(x)=5\), which statement about its graph is correct?MediumLevel 23If p(x) is the zero polynomial, which statement about its graph is correct?MediumLevel 23If \(p(2)=0\), \(p(5)=0\) and \(p(9)=0\), how many distinct intersections with the x-axis will the graph have?MediumLevel 23If a parabola cuts the (x)-axis at ((-2,0)) and ((6,0)), through which (x)-value can its axis of symmetry pass?MediumLevel 23If (p(x)=x^2+2x+5), which statement about its real zeroes is correct?MediumLevel 23On the graph of a function we have \(p(3)=0\) and \(p(7)=4\). Which statement is correct?MediumLevel 23If p(x) = 2x^2 - 8, what are the zeros of p(x)?MediumLevel 23A graph has an (x)-axis intersection ((r,0)). If (r<0), what can be said about the zero?MediumLevel 23If (p(x)=-(x+3)^2), how will the graph meet the (x)-axis?MediumLevel 23If a graph (not just a removable point) intersects the x-axis at (-1,0), (2,0) and (4,0), what is the product of its zeros?MediumLevel 23