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
Which of the following points shows that x = 9 is a zero of the polynomial p(x)?MediumLevel 24If a graph contains the points \\((2,3)\\) and \\((5,0)\\), which x-value is definitely a zero of the polynomial/function?MediumLevel 24If p(x) = x² + x − 6, what are its real zeroes?MediumLevel 24A graph cuts the x-axis at (a, 0), (0, 0) and (-a, 0) where \(a \neq 0\). What is the sum of the zeroes?MediumLevel 24If (p(3)<0), (p(4)=0), (p(5)<0), at which given (x)-value will the graph meet the (x)-axis?MediumLevel 24The graph of a polynomial crosses the x-axis at two distinct points and does not merely touch it at any point. What does this imply?MediumLevel 24If (p(x)=x^2+1), why does its graph not cut the (x)-axis?MediumLevel 24If a graph intersects the x-axis at (-5, 0) and (5, 0), what is the product of those zeroes?MediumLevel 24If \(p(x)=3x-15\), what is the x-axis intersection (x-intercept) of its graph?MediumLevel 24If the graph of a polynomial touches the x-axis at the single point (m, 0) and does not meet the x-axis anywhere else, how many distinct real zeros does the polynomial have?MediumLevel 24If (p(x)=(x-2)(x-5)(x-8)), at how many distinct points will the graph cut the (x)-axis?MediumLevel 24If the y-axis intercept of a graph is (0, 0), what additional conclusion is correct?MediumLevel 24If p(x) = x² − 12x + 36, what is the distinct real zero?MediumLevel 24If a graph cuts the x-axis at (−2, 0), (3, 0), and (7, 0), what is the product of the zeroes?MediumLevel 24If (p(x)=4), what is correct about the solutions of (p(x)=0)?MediumLevel 24If p(x) = 0 is the zero polynomial, which statement about its zeroes is correct?MediumLevel 24If a graph cuts the x-axis at the points (2,0) and (14,0), what is the average of the zeroes?MediumLevel 24If \(p(x)=-(x-1)(x+4)\), at which real zeros (x-intercepts) does its graph cross the x-axis?MediumLevel 24If p(−6) = 0, p(0) = 0, and p(5) = 0, which statement about the graph is correct?MediumLevel 24If (p(x)=(x+3)(x-1)^2), at how many distinct points will the graph meet the (x)-axis?MediumLevel 24