Polynomial | A unique presentation of geometrical meaning of the zeroes of a polynomial | Part-1

Geometrical Meaning of the Zeroes of a Polynomial

What is a Polynomial?

A polynomial is a mathematical expression that involves variables, constants, and the operations of addition, subtraction, and multiplication. It is written in the form: P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, where a_n, a_n-1, ..., a₀ are real numbers and n is a non-negative integer.

Classification of Polynomials Based on Number of Terms

Polynomials can be grouped according to how many terms they contain:

• Monomial: A polynomial with one term (e.g., 5x)
• Binomial: An expression made up of exactly two distinct terms (e.g., x + 3)
• Trinomial: A polynomial with three terms (e.g., x² + 2x + 1)
• Polynomial: A mathematical expression consisting of one or more terms combined together

Names of a Polynomial According to Degree

The degree of a polynomial refers to the largest exponent of the variable present in the expression. Based on degree, polynomials are named as:

• Constant Polynomial: Degree 0 (e.g., 4)
• Linear Polynomial: Degree 1 (e.g., 2x + 3)
• Quadratic Polynomial: A polynomial where the highest power of the variable is 2 (for example, x² - 5x + 6)
• Cubic Polynomial: Degree 3 (e.g., x³ - 3x² + 2)
• Quartic Polynomial: Degree 4 (e.g., x⁴ + 2x³ - x + 1)

The Value of a Polynomial

The value of a polynomial for a particular value of the variable is obtained by substituting that value into the expression.
As an illustration, consider P(x) = x² + 2x + 1 when x = 2P(2) = 2² + 2×2 + 1 = 4 + 4 + 1 = 9.

What is the Zero of a Polynomial?

A zero (also called a root) of a polynomial is a value of the variable for which the polynomial evaluates to zero. In other words, if P(a) = 0, then a is a zero of the polynomial P(x).

Geometrical Meaning of the Zeroes of a Polynomial

The geometric interpretation of a polynomial's zeroes becomes clear when its graph is plotted on the Cartesian coordinate plane.

For a polynomial P(x), its graph is a curve. The zeroes of the polynomial are the x-coordinates of the points where the graph intersects the x-axis.

• The graph of a linear polynomial appears as a straight line. It intersects the x-axis at one point, which represents its single zero.
• For a quadratic polynomial, the graph is a parabola. It can intersect the x-axis at 0, 1, or 2 points, representing the number of real zeroes.
• For a cubic polynomial, the graph may intersect the x-axis at one, two, or three points, corresponding to up to three real zeroes.

Hence, geometrically, zeroes of a polynomial are the x-values where the curve of the polynomial touches or crosses the x-axis.

Geometrical Meaning When P(x) is a Linear Polynomial

A linear polynomial has the general form P(x) = ax + b, with a ≠ 0. Its graph represents a straight line on the coordinate plane.

The zero of the polynomial is the value of x for which P(x) = 0. This happens when:

ax + b = 0 ⇒ x = -b/a

Geometrically, this means the line y = ax + b intersects the x-axis at the point (-b/a, 0). This point is the zero of the polynomial.

Since the x-axis represents all points where y = 0, any intersection with the x-axis corresponds to a root or zero of the polynomial. For a linear polynomial, this intersection occurs exactly once.