Unit 2: Polynomial

Introduction to Polynomial

An algebraic expression in which the exponent of each variable is a whole number is called a polynomial. 

Example: \(2x^{\frac{1}{2}}\) (not polynomial), \(2x^{-2}\) (not polynomial), \(2x^{2}\) (polynomial)

The general form of a n degree Polynomial can be written as.

Where, 

1. \(a_n, a_{n-1}, a_{n-2}, ….. ….. a_1, a_0 \) are are real number coefficients.
2. n  is a whole number (the degree of the polynomial)
3. The exponent of the variable x  must be either zero or a positive whole number only

Key Properties

• Since a polynomial is a function, it is used not only in mathematics but also in practical applications such as demand, supply, velocity, acceleration, cost, production, and other fields.
• Polynomials can be classified by degree: Linear (degree 1), Quadratic (degree 2), Cubic (degree 3), Quartic (degree 4), etc.

Examples: \(f(x) = x^4 – 3x^3 + 5x^2 + 4\) (4 degree polynomial), \(x^3 + 2x^2 + 3x - 6\) (3 degree polynomial

Division Theorem

Statement:

Proof

When the polynomial $f(x)$ is divided by the divisor $(x - a)$, there exist the quotient $q(x)$ and the remainder $R$. Then, according to the ordinary division method, the relation among $f(x)$, $(x - a)$, $q(x)$, and $R$ is:

$f(x) = q(x) \times (x - a) + R \quad \ldots (i)$

Substituting $x = a$ in equation $(i)$:

$f(a) = q(a) \times (a - a) + R$

$f(a) = q(a) \times 0 + R$

$\boxed{f(a) = R}$

Therefore, $R = f(a)$. Hence, proved. $\blacksquare$

Division Relation

If $f(x)$ is divided by $d(x)$ with quotient $q(x)$ and remainder $R$:

$f(x) = d(x) \times q(x) + R$

Or equivalently:

$\frac{f(x)}{d(x)} = q(x) + \frac{R}{d(x)}$

If $R = 0$, then $\frac{f(x)}{d(x)} = q(x)$, meaning $f(x)$ is exactly divisible by $d(x)$.

Rational Root Theorem

The Rational Root Theorem (sometimes called the Rational Zero Theorem) is a powerful tool in algebra. It helps you find the possible rational solutions (roots) of a polynomial equation with integer coefficients. For Class 10 students, this theorem is especially useful for solving polynomial equations of **degree 3 or higher**, which cannot be easily factored using simpler methods.

Statement

Step-by-Step Method to Find Rational Roots

1.  **Identify the constant term (\(a_0\)) and the leading coefficient (\(a_n\))** of the polynomial.

2.  **List all factors of the constant term.** These are the possible values for the numerator \(p\).

3.  **List all factors of the leading coefficient.** These are the possible values for the denominator \(q\).

4.  **Form all possible fractions \(\pm \frac{p}{q}\)** and simplify. Remove any duplicates. This gives you the complete list of **possible rational roots**.

5.  **Test each possible root** by substituting it into the polynomial. You can use direct substitution or **synthetic division**. If the result is zero, you have found a root.