Degree of a Polynomial for CBSE Class 10 Mathematics

In algebra, the degree of a polynomial is its single most informative characteristic. It dictates how the polynomial behaves when graphed, how rapidly its value grows as the variable increases, and sets the strict mathematical upper bound on the number of zeroes the polynomial can possess.

In CBSE Class 10 Mathematics, finding the degree of a polynomial is tested directly in one-mark questions and serves as an essential prerequisite for factoring, polynomial division, and solving quadratic equations.

What You Will Learn

• Formal definition of the degree of a single-variable polynomial
• The rule of the highest non-zero exponent
• Degree of special polynomials: constant polynomials vs. the zero polynomial
• Rules governing degree under algebraic operations (addition, subtraction, multiplication, division)
• Finding the degree of unsimplified or bracketed algebraic expressions
• Solved CBSE board examination problems and common traps

1. Definition of Degree of a Polynomial

Formal Statement

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

For a polynomial in standard form: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0, \quad \text{with } a_n \ne 0$ the degree of $P(x)$ is $n$.

Quick Examples

1. In $P(x) = 7x^4 - 3x^2 + 2x - 9$:
   • The powers of $x$ present are $4, 2, 1,$ and $0$.
   • The highest power is $4$.
   • Therefore, the degree is $4$.
2. In $Q(y) = 5 - 2y + 8y^3$:
   • The powers of $y$ are $0, 1,$ and $3$.
   • The highest power is $3$.
   • Therefore, the degree is $3$.

Important: <u>The degree must always be determined after the polynomial has been fully expanded and all like terms have been combined.</u>

2. Degree of Special Polynomials

1. Non-Zero Constant Polynomial

A polynomial containing only a single non-zero number, such as $P(x) = 8$ or $Q(x) = -15$:

• Can be rewritten as $P(x) = 8x^0$.
• The exponent of the variable is $0$.
• Therefore, the degree of any non-zero constant polynomial is $0$.

2. The Zero Polynomial

The polynomial $P(x) = 0$:

• Can be written as $0 \cdot x^0, 0 \cdot x^1, 0 \cdot x^2, 0 \cdot x^{100}, \dots$.
• Any non-negative integer could be considered an exponent, but every coefficient is $0$.
• By mathematical convention, the degree of the zero polynomial is NOT defined (undefined).

Remember:

• Constant polynomial $c \ne 0 \implies \text{Degree } = 0$.
• Zero polynomial $0 \implies \text{Degree is undefined}$.

3. Degree Under Algebraic Operations

When combining polynomials through arithmetic operations, their degrees follow strict algebraic rules:

Let $P(x)$ and $Q(x)$ be non-zero polynomials with $\deg(P) = m$ and $\deg(Q) = n$.

1. Addition and Subtraction

$\deg[P(x) \pm Q(x)] \le \max(m, n)$

• If $m \ne n$, then $\deg[P \pm Q] = \max(m, n)$.
• If $m = n$, the leading terms may cancel out, causing the degree of the result to be strictly less than $m$. Example: If $P(x) = x^3 + 2x$ and $Q(x) = x^3 - 5$, then $P(x) - Q(x) = 2x + 5$, which has degree $1 < 3$.

2. Multiplication

$\deg[P(x) \cdot Q(x)] = \deg[P(x)] + \deg[Q(x)] = m + n$ When two polynomials are multiplied, their highest exponents add together ($x^m \cdot x^n = x^{m+n}$). Example: $(2x^2 + 1)(3x^3 - 4) \implies$ Degree is $2 + 3 = 5$.

3. Division

If $P(x)$ is divided by non-zero $Q(x)$ such that $P(x) = Q(x) \cdot q(x) + r(x)$: $\deg[q(x)] = \deg[P(x)] - \deg[Q(x)] = m - n \quad (\text{provided } m \ge n)$ and for the remainder: $r(x) = 0 \quad \text{or} \quad \deg[r(x)] < \deg[Q(x)]$

4. Solved CBSE Board Examination Problems

Solved Example 1: Degree of an Expanded Expression

Problem: Find the degree of the polynomial $P(x) = (x + 1)(x^2 - x + 1) - (x^3 - 4x + 2)$.

Solution:

1. Expand the first product using the sum of cubes algebraic identity $(a+b)(a^2-ab+b^2) = a^3+b^3$: $(x + 1)(x^2 - x + 1) = x^3 + 1$
2. Substitute into $P(x)$: $P(x) = (x^3 + 1) - (x^3 - 4x + 2)$
3. Simplify by combining like terms: $P(x) = x^3 + 1 - x^3 + 4x - 2 = 4x - 1$
4. In $4x - 1$, the highest power of $x$ is $1$.
5. Therefore, <u>the degree of the polynomial is $1$</u>.

Solved Example 2: Degree of a Power Expression

Problem: What is the degree of the polynomial $f(x) = (x^2 + 2)^3 - x^6$?

Solution:

1. Expand $(x^2 + 2)^3$ using the binomial formula $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$: $(x^2 + 2)^3 = (x^2)^3 + 3(x^2)^2(2) + 3(x^2)(2^2) + 2^3 = x^6 + 6x^4 + 12x^2 + 8$
2. Substitute back into $f(x)$: $f(x) = (x^6 + 6x^4 + 12x^2 + 8) - x^6$ $f(x) = 6x^4 + 12x^2 + 8$
3. The highest power of $x$ with a non-zero coefficient is $4$.
4. Therefore, <u>the degree of $f(x)$ is $4$</u>.

5. Summary and Examination Tips

Exam Tip: Whenever you encounter bracketed polynomial expressions in board questions, never guess the degree by looking at individual terms. Always multiply or expand and check whether leading terms cancel out!

Common Mistake: Stating that the degree of $P(x) = 5$ is 1 because there is "one number". A number without a variable represents $x^0$, making its degree $0$.