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 polynomial meets the x-axis at (a, 0) and (−a, 0). If a ≠ 0, what is the sum of its zeros?MediumLevel 23If the real zeroes of a polynomial are written as 2, 2 and −5, what are the distinct real zeroes?EasyLevel 23If p(x) = x³ − x, at which x-values can the graph meet the x-axis?MediumLevel 23If a graph touches the \(x\)-axis only at \(x=11\), which statement is correct?MediumLevel 23If p(x) = x² − 7x + 12, what are the x-axis intersections of the graph?MediumLevel 23If a graph has x-axis intersections at (3, 0) and (12, 0), what is the average of its zeros?MediumLevel 23A polynomial graph cuts the x-axis at x = −5 and x = 5. What does this indicate with respect to the y-axis?EasyLevel 23If \(p(1)<0\), \(p(2)=0\) and \(p(3)>0\), which of the following points lies on the \(x\)-axis?MediumLevel 23If a graph cuts the (x)-axis at ((-6,0)), ((-2,0)), ((4,0)), which is the greatest zero?MediumLevel 23If (p(x)=(x-1)(x-4)(x+2)), at how many distinct points will the graph cut the (x)-axis?MediumLevel 23If the graph of a polynomial passes through the points \((-4,0), (0,3), (2,0)\) and \((5,-1)\), which are its real zeroes?MediumLevel 24A parabola cuts the x-axis at x = -5 and x = 3. What is the average of its zeroes?MediumLevel 24If p(x) = (x + 2)(x − 7), at which points will the graph meet the x-axis?EasyLevel 24If the graph of a polynomial crosses the x-axis at \(x=1\) and only touches the x-axis at \(x=6\), how many distinct real zeros does the polynomial have?MediumLevel 24A polynomial graph cuts the (x)-axis at ((-9,0)), ((-1,0)) and ((4,0)). Which is the greatest zero?MediumLevel 24If \(p(-3)=0\), \(p(2)=5\) and \(p(8)=0\), at which points does the graph intersect the x-axis?MediumLevel 24If \(p(x)=x^2-10x+25\), how does its graph meet the x-axis?MediumLevel 24If p(x) = x² + 6x + 10, which statement about its real zeroes is correct?MediumLevel 24A student saw the point (0, −6) and wrote −6 as a zero (root) of a function. Why is this incorrect?MediumLevel 24If the zeroes of a polynomial's graph are -4 and 9, what is the distance between them on the x-axis?MediumLevel 24