Degree of a Polynomial

The degree of a polynomial in one variable is the highest power (exponent) of the variable that appears in the polynomial with a non-zero coefficient.

For a polynomial p(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n is not zero, the degree is n.

Examples:

p(x) = 4x^5 - 3x^2 + 7x - 1: Degree = 5 (the highest power is x^5).

q(x) = 2x^2 + 9x + 4: Degree = 2 (the highest power is x^2).

r(x) = 6x - 11: Degree = 1 (the highest power is x^1).

s(x) = 8: Degree = 0 (a non-zero constant has degree 0; think of 8 as 8x^0).

Special cases:

Zero polynomial: p(x) = 0. All coefficients are zero. The degree of the zero polynomial is not defined (or sometimes stated as negative infinity). This is because no term with a non-zero coefficient exists.

Constant polynomial (non-zero): p(x) = c, where c is not 0. The degree is 0.

Important notes:

1. The degree depends on the highest power with a non-zero coefficient. If p(x) = 0x^3 + 5x^2 + 1, the degree is 2 (not 3, because the coefficient of x^3 is 0).

2. The terms do not need to be in descending order. For p(x) = 3 + 2x - x^4, the degree is still 4.

3. For polynomials in more than one variable (multivariable), the degree is the highest sum of exponents in any single term. For example, in 3x^2y^3 + 2xy, the degrees of terms are 2+3=5 and 1+1=2, so the polynomial has degree 5. However, in Class 9, you mostly study polynomials in one variable.

Why does a polynomial of degree n have at most n zeroes?

This is an important theorem, and here is an intuitive explanation suitable for Class 9 students.

Observation for small degrees:

Degree 1 (Linear): p(x) = ax + b. Setting p(x) = 0: ax + b = 0, so x = -b/a. Exactly one zero. A straight line crosses the x-axis at most once.

Degree 2 (Quadratic): p(x) = ax^2 + bx + c. The quadratic formula gives at most 2 solutions. A parabola can cross the x-axis at 0, 1, or 2 points.

Degree 3 (Cubic): A cubic polynomial can be factored as a(x - r)(x^2 + ...) or a(x - r1)(x - r2)(x - r3), giving at most 3 zeroes.

General principle (Factor Theorem connection):

Step 1: If r is a zero of p(x), then by the Factor Theorem, (x - r) is a factor of p(x).

Step 2: So p(x) = (x - r) x q(x), where q(x) has degree (n - 1).

Step 3: If p(x) has another zero s (different from r), then s must be a zero of q(x), giving another factor (x - s), reducing the degree further.

Step 4: Each new zero reduces the remaining degree by 1. Starting from degree n, we can find at most n factors (x - r1), (x - r2), ..., (x - rn).

Step 5: Therefore, a polynomial of degree n has at most n zeroes.

Note: The polynomial may have fewer than n real zeroes. For example, x^2 + 1 has degree 2 but zero real zeroes (since x^2 + 1 > 0 for all real x).

Polynomials are classified by their degree into the following standard categories:

1. Constant Polynomial (Degree 0):

A polynomial with no variable term: p(x) = c, where c is a non-zero real number.

Examples: p(x) = 5, q(x) = -3, r(x) = sqrt(2).

Graph: A horizontal line at height c. It never crosses the x-axis (if c is not 0), so it has zero zeroes.

2. Linear Polynomial (Degree 1):

General form: p(x) = ax + b, where a is not zero.

Examples: p(x) = 3x + 7, q(x) = -x + 4, r(x) = (2/3)x.

Graph: A straight line. It has exactly one zero: x = -b/a.

The name 'linear' comes from the fact that its graph is a line.

3. Quadratic Polynomial (Degree 2):

General form: p(x) = ax^2 + bx + c, where a is not zero.

Examples: p(x) = x^2 - 4x + 3, q(x) = 2x^2 + 1, r(x) = -x^2 + 5x.

Graph: A parabola (U-shaped or inverted U). It can have 0, 1, or 2 real zeroes.

The word 'quadratic' comes from 'quad' meaning square (the variable is squared).

4. Cubic Polynomial (Degree 3):

General form: p(x) = ax^3 + bx^2 + cx + d, where a is not zero.

Examples: p(x) = x^3 - 6x^2 + 11x - 6, q(x) = 2x^3 + 1.

Graph: An S-shaped curve. It can have 1, 2, or 3 real zeroes. A cubic always has at least 1 real zero.

5. Biquadratic (Quartic) Polynomial (Degree 4):

General form: p(x) = ax^4 + bx^3 + cx^2 + dx + e, where a is not zero.

Examples: p(x) = x^4 - 5x^2 + 4, q(x) = x^4 + 1.

Up to 4 real zeroes.

6. Higher-degree polynomials (degree 5 and above) are called by their degree: quintic (5), sextic (6), etc.

The degree of a polynomial plays a central role in mathematics and its applications:

Curve Fitting: When scientists collect data points and want to find an equation that fits them, the degree of the polynomial chosen determines the shape of the curve. A degree-1 polynomial gives a straight line (linear regression), degree-2 gives a parabola (quadratic regression), and so on.

Graph Behaviour: The degree tells you how the polynomial graph behaves. Linear polynomials produce lines, quadratics produce parabolas, and cubics produce S-curves. The degree also determines how many times the graph can cross the x-axis.

Solving Equations: The degree determines the maximum number of solutions to the equation p(x) = 0. A degree-3 equation can have up to 3 solutions, which is critical in engineering, physics, and optimisation.

Complexity in Computing: Algorithm efficiency is often expressed as a polynomial in the input size: O(n), O(n^2), O(n^3). The degree of this polynomial tells you how the running time grows — linear (fast), quadratic (moderate), or cubic (slow).

Physics — Projectile Motion: The height of a projectile is a quadratic polynomial in time (degree 2): h = -gt^2/2 + v_0t + h_0. This determines the parabolic trajectory.