NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Exercise 2.1

The Geometric Meaning of Polynomial Zeroes

In algebra, a real number $k$ is called a zero of a polynomial $p(x)$ if $p(k) = 0$. Geometrically, this has a highly intuitive meaning. If we plot the graph of the equation $y = p(x)$ on a Cartesian coordinate plane, the zeroes of the polynomial are the x-coordinates of the points where the graph intersects or touches the x-axis.

Why does this happen? At any point on the x-axis, the y-coordinate is strictly zero. Since our graph represents $y = p(x)$, setting $y = 0$ is mathematically equivalent to solving the algebraic equation $p(x) = 0$. Therefore, every single point where the curve meets the x-axis represents a real value of $x$ that makes the polynomial equal to zero. For a linear polynomial $ax + b$ ($a \neq 0$), the graph is a straight line intersecting the x-axis at exactly one point, $(-b/a, 0)$. For a quadratic polynomial $ax^2 + bx + c$, the graph is a parabolic curve that can cross the x-axis at two points, touch it at one point, or completely miss it, representing two, one, or zero real zeroes respectively.

Step-by-Step: Counting Zeroes from a Graph

1. Identify the Dependent Variable — Read the problem statement carefully to confirm if the function is defined as $y = p(x)$ or $x = p(y)$. For standard Class 10 Exercise 2.1 problems, we always analyze $y = p(x)$.
2. Locate the Horizontal X-Axis — Focus your eyes strictly on the horizontal axis (the x-axis). You must completely ignore any points where the graph intersects the vertical y-axis.
3. Trace the Curve — Follow the path of the given graph from left to right along the Cartesian plane.
4. Count Intersections and Touchpoints — Count every distinct point where the curve cuts directly across the x-axis or gently touches the x-axis and reverses its direction.
5. State the Final Number of Zeroes — The total number of these intersection and touching points is exactly equal to the number of real zeroes of the polynomial $p(x)$.

Beware of the Variable Swap Trap!

In board exams, examiners love to test your conceptual depth by switching the standard variables. While class 10 maths polynomials ex 2 1 exclusively deals with the polynomial equation $y = p(x)$, a tricky question might ask you to find the number of zeroes of a polynomial defined as $x = p(y)$.

If the equation is written as $x = p(y)$, the zeroes are the values of $y$ for which $x = 0$. In this specific scenario, you must count the number of times the curve intersects or touches the vertical y-axis, not the horizontal x-axis! Always read the notation carefully before writing your final answer.

Practice Questions with Solutions

• Q: The graph of a linear polynomial $y = p(x)$ is a straight line parallel to the x-axis, passing through the point $(0, 4)$. Find the number of zeroes of $p(x)$. A: Step 1: Analyze the given line. A straight line parallel to the x-axis passing through $(0, 4)$ is represented by the equation $y = 4$. Step 2: Check for intersections with the x-axis. Since the line is completely parallel to the x-axis and lies 4 units above it, it will never cross or touch the x-axis. Step 3: Relate this to zeroes. Since there are 0 points of intersection with the x-axis, the polynomial has no real zeroes. Final answer: The number of zeroes of $p(x)$ is 0.
• Q: A parabola opening upwards cuts the horizontal x-axis at the points $(-3, 0)$ and $(5, 0)$. How many zeroes does this quadratic polynomial have, and what are their values? A: Step 1: Identify all points of intersection with the x-axis. The curve intersects the x-axis at $(-3, 0)$ and $(5, 0)$. Step 2: Count the total number of intersection points, which is exactly 2. Step 3: Determine the zeroes. The x-coordinates of these intersection points are the zeroes of the polynomial. Thus, the zeroes are $x = -3$ and $x = 5$. Final answer: The number of zeroes is 2, and the zeroes are -3 and 5.
• Q: The graph of a cubic polynomial $y = p(x)$ crosses the origin $(0,0)$, passes through $(-2, 0)$, and touches the x-axis at $(3, 0)$ before turning upwards. Find the number of zeroes of $p(x)$. A: Step 1: Identify all coordinate points where the graph meets the horizontal x-axis. The points are $(-2, 0)$, $(0, 0)$ (the origin), and $(3, 0)$. Step 2: Analyze the contact. The graph cuts across at $(-2, 0)$ and $(0, 0)$, and touches at $(3, 0)$. Both crossings and touchpoints count as geometric zeroes. Step 3: Sum up the points: $1 + 1 + 1 = 3$ points of contact. Final answer: The number of zeroes of $p(x)$ is 3.
• Q: If a curve defined by $y = p(x)$ is positioned entirely below the x-axis and never rises above $y = -1$, find the number of real zeroes of the polynomial $p(x)$. A: Step 1: Visualize the position of the graph. The maximum value of $y$ on the curve is $-1$, which means the entire curve lies in the third and fourth quadrants. Step 2: Check for any contact with the horizontal axis. Since the curve stays below $y = -1$, it cannot touch or intersect the line $y = 0$ (the x-axis). Step 3: Conclude the number of zeroes. With zero intersections, there are no real zeroes. Final answer: The number of zeroes of $p(x)$ is 0.

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