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 the graph of a polynomial \(p(x)\) intersects the x-axis at the point \((r,0)\), which of the following statements is always true?ExpertLevel 24At which x-values will the graph of the quadratic p(x)=x^2+2x-15 intersect the x-axis?ExpertLevel 24For a polynomial p(x) we are given p(-3)<0, p(1)=0 and p(4)>0. Which zero is certainly a root?ExpertLevel 24The graph does not cut the x-axis but cuts the line y = 2 twice. What is certain about its zeroes?MediumLevel 24If p(x)=4x, at which point does its graph intersect the x-axis?ExpertLevel 24If a quadratic graph cuts the (x)-axis at ((m,0)) and ((n,0)), which polynomial can have the same zeroes?ExpertLevel 24If a graph crosses the x-axis at 2.5, what is the zero (root) of the polynomial?ExpertLevel 24If \(p(x)=x^2-2x-8\), at which points does its graph intersect the x-axis?ExpertLevel 24A polynomial graph cuts the (x)-axis only at the even numbers (-4), (2), and (8). What is the number of zeroes?ExpertLevel 24If the graph of a polynomial touches the (x)-axis at \(\frac{1}{2}\), which statement is correct?ExpertLevel 24If the graph of a polynomial intersects the x-axis at \((-\sqrt{3},0)\), what is the zero (root) of the polynomial?ExpertLevel 24Which statement is geometrically incorrect?ExpertLevel 24If \(p(x)=x^2+6x+10\), how does its graph relate to the x-axis?ExpertLevel 24If \(p(x)=2x^2-8\), at which \(x\)-values will the graph cut the \(x\)-axis?ExpertLevel 24If a graph cuts the x-axis at -7 and -2, what are the solutions of p(x) = 0?EasyLevel 24A quadratic polynomial graph does not touch or intersect the x-axis. What is the number of distinct real zeroes?MediumLevel 24If p(x) = 5(x + 1)(x - 2)(x - 4), at how many distinct points will the graph cut the x-axis?MediumLevel 24If the graph of a polynomial goes from above the x-axis to below it and the point (9,0) lies exactly at that crossing, what does (9,0) represent?ExpertLevel 24If the graph of \(p(x)=x^2-11x+30\) is drawn, at which points does it intersect the x-axis?ExpertLevel 24If a quadratic polynomial graph may have (0), (1), or (2) intersection points with the (x)-axis, which statement is most correct?ExpertLevel 24