Exercise 3.2 — Graphical Representation

Connecting Graphs and Zeroes of a Polynomial

Exercise 3.2 of the Class 10 Polynomials chapter explores the relationship between a polynomial and its graph. The central idea is that the zeroes of a polynomial are exactly the x-coordinates of the points where its graph crosses or touches the x-axis. This visual connection helps students understand why polynomials of different degrees can have different numbers of zeroes.

For a linear polynomial like p(x) = 2x − 5, setting p(x) = 0 gives x = 5/2, which is also the point where the straight line y = 2x − 5 crosses the x-axis. This shows that a linear polynomial ax + b, where a is not zero, always has exactly one zero, corresponding to a single intersection point with the x-axis.

How Quadratic Polynomials Behave on a Graph

The graph of a quadratic polynomial ax² + bx + c is a curve called a parabola. If the coefficient a is positive, the parabola opens upward; if a is negative, it opens downward. The number of zeroes of a quadratic polynomial depends on how many times this parabola meets the x-axis, and there are three possible situations.

• Two distinct zeroes — the parabola cuts the x-axis at two different points
• One zero (repeated) — the parabola touches the x-axis at exactly one point, meaning the two zeroes coincide
• No real zeroes — the parabola lies entirely above or entirely below the x-axis and never touches it

For example, the quadratic polynomial x² − 5x + 6 factorizes as (x − 2)(x − 3), giving zeroes at x = 2 and x = 3. These values match exactly with the points where the graph of y = x² − 5x + 6 intersects the x-axis, confirming the connection between algebra and geometry.

Worked Example: Finding Zeroes by Factorization

To find the zeroes of a polynomial algebraically, set p(x) = 0 and solve the resulting equation. For p(x) = x² + 5x + 6, splitting the middle term gives x² + 3x + 2x + 6 = 0, which factorizes to (x + 3)(x + 2) = 0. Setting each factor to zero gives x = −3 and x = −2, which are the two zeroes of this polynomial.

For higher-degree polynomials such as p(x) = x⁴ − 16, the expression can be factorized step by step using the difference of squares: x⁴ − 16 = (x² − 4)(x² + 4) = (x − 2)(x + 2)(x² + 4). Since x² + 4 can never equal zero for real values of x, the only real zeroes of this polynomial are x = 2 and x = −2.

Plotting a Parabola Point by Point

A key skill in this exercise is drawing the graph of a quadratic polynomial by first creating a table of values. For p(x) = x² − x − 12, substituting values of x from −4 to 5 gives a set of (x, y) points. Plotting these points and joining them with a smooth curve produces a parabola that crosses the x-axis at x = −3 and x = 4 — and these are exactly the zeroes obtained by factorizing x² − x − 12 as (x + 3)(x − 4).

This graphical method is especially useful for verifying algebraic answers. After finding the zeroes by factorization, students can substitute these values back into the polynomial to confirm that p(x) = 0, which is called justification and is often required in board exam answers.

Cubic Polynomials and Their Graphs

Cubic polynomials, which have degree 3, can have at most three zeroes, and their graphs can intersect the x-axis at one, two, or three points depending on the polynomial. For example, the graph of y = x³ − 4x crosses the x-axis at three distinct points, giving three real zeroes, while the graph of y = x³ touches the x-axis at only the origin, giving a single zero.

This pattern follows a general rule that students should remember clearly for board exams: a linear polynomial has at most one zero, a quadratic polynomial has at most two zeroes, and a cubic polynomial has at most three zeroes. The degree of the polynomial sets the maximum possible number of zeroes.

Common Mistakes to Avoid

• Forgetting that a parabola touching the x-axis at one point still represents a quadratic with a repeated zero, not "no zero"
• Assuming every quadratic polynomial must have real zeroes — some parabolas never touch the x-axis
• Errors in splitting the middle term while factorizing, especially with negative coefficients
• Not verifying answers by substituting the zero back into the original polynomial
• Confusing the degree of a polynomial with the maximum number of zeroes when graphing higher-degree expressions

What This Lesson Prepares You For

Understanding the graphical behavior of polynomials and the link between zeroes and factorization sets the stage for the next topics in this chapter, including the relationship between zeroes and coefficients, and the division algorithm for polynomials. These graphing and factorization skills are also directly useful when studying quadratic equations and coordinate geometry.

For a refresher on the basic definitions of degree, value, and zeroes used here, revisit Exercise 3.1 and the introduction to polynomials.