Polynomials – Class 9 Mathematics

1. Definition of a Polynomial

A polynomial is an algebraic expression consisting of variables and coefficients, involving only non-negative integer powers of variables.

General form of a polynomial in one variable $x$:

$$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$

where:

• $a_0, a_1, \dots, a_n$ are constants called coefficients.

• $n$ is a non-negative integer, called the degree of the polynomial.

• $a_n \neq 0$.

2. Types of Polynomials

1. Monomial: A polynomial with one term
   Example: $5x^3$

2. Binomial: A polynomial with two terms
   Example: $x^2 + 3x$

3. Trinomial: A polynomial with three terms
   Example: $x^2 + x + 1$

4. Zero Polynomial: A polynomial in which all coefficients are 0
   Example: $0$

   • Degree of zero polynomial: not defined

3. Degree of a Polynomial

• The degree is the highest power of the variable in the polynomial.

• Examples:

  • $7x^4 + 3x^3 – x + 2$ → Degree = 4

  • $5x – 9$ → Degree = 1

4. Coefficients

• Coefficient of a term: numerical factor of the term

• Example: In $3x^2 + 4x + 7$

  • Coefficient of $x^2$ = 3

  • Coefficient of $x$ = 4

  • Constant term = 7

5. Types of Polynomials Based on Degree

6. Zeros of a Polynomial

• A number $\alpha$ is called a zero of the polynomial $P(x)$ if

$$P(\alpha) = 0$$

• Example: For $P(x) = x^2 – 5x + 6$
  Solve $x^2 – 5x + 6 = 0 \Rightarrow x = 2, 3$
  So, 2 and 3 are zeros of $P(x)$.

7. Relation Between Zeros and Coefficients

For a quadratic polynomial $ax^2 + bx + c$:

• Sum of zeros:

$$\alpha + \beta = -\frac{b}{a}$$

• Product of zeros:

$$\alpha \beta = \frac{c}{a}$$

8. Division Algorithm for Polynomials

• If $P(x)$ and $g(x)$ are polynomials, $g(x) \neq 0$, then

$$P(x) = g(x) \cdot Q(x) + R(x)$$

where:

• $Q(x)$ = quotient

• $R(x)$ = remainder, with degree $\deg(R) < \deg(g)$

• Example: Divide $x^3 – 3x^2 + 5$ by $x – 2$

$$x^3 – 3x^2 + 5 = (x – 2)(x^2 – x – 2) + 1$$

9. Factor Theorem

• If $P(\alpha) = 0$, then $(x – \alpha)$ is a factor of $P(x)$.

• Conversely, if $(x – \alpha)$ is a factor, then $P(\alpha) = 0$.

Example: $P(x) = x^2 – 5x + 6$

• $P(2) = 2^2 – 5(2) + 6 = 0 \Rightarrow (x-2)$ is a factor

• $P(3) = 3^2 – 15 + 6 = 0 \Rightarrow (x-3)$ is a factor

10. Remainder Theorem

• The remainder when a polynomial $P(x)$ is divided by $(x – a)$ is

$$R = P(a)$$

Example: Divide $x^3 – 4x^2 + 5x – 2$ by $x – 1$

• Remainder = $P(1) = 1 – 4 + 5 – 2 = 0$ → divisible

11. Graphs of Polynomials

• Graph of a polynomial is a smooth curve (no breaks).

• Degree determines shape and number of turning points:

  • Linear: straight line

  • Quadratic: parabola

  • Cubic: S-shaped curve

12. Important Points

• Polynomial expressions cannot have negative or fractional powers.

• A polynomial in one variable of degree $n$ has at most $n$ zeros.

• Factorisation and zeros are closely related: knowing zeros → easy factorisation.

Example Problems

1. Find the zeros of $P(x) = x^2 – 7x + 12$.

2. Verify the sum and product of zeros for $x^2 – 5x + 6$.

3. Divide $2x^3 + 3x^2 – x + 5$ by $x + 2$.

4. Determine if $x – 3$ is a factor of $x^3 – 7x + 6$.