Polynomials Class 10 Mathematics Concept notes and Formula Sheet

 

Concept Notes

1. Definition of a Polynomial

• A polynomial is an algebraic expression consisting of variables (or indeterminates) raised to non-negative integer powers and combined using addition, subtraction, and multiplication. The general form of a polynomial is: $$P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$ where $a_n, a_{n-1}, \ldots, a_0$ are constants (coefficients), and $n$ is a non-negative integer representing the degree of the polynomial.

2. Degree of a Polynomial

• The degree of a polynomial is the highest power of the variable in the polynomial. For example, in $3x^4 - 5x^2 + 2$, the degree is 4.

3. Types of Polynomials

• Constant Polynomial: A polynomial of degree 0. Example: $P(x) = 5$.
• Linear Polynomial: A polynomial of degree 1. Example: $P(x) = 2x + 3$.
• Quadratic Polynomial: A polynomial of degree 2. Example: $P(x) = x^2 - 4x + 4$.
• Cubic Polynomial: A polynomial of degree 3. Example: $P(x) = x^3 - 3x^2 + 3x - 1$.

4. Polynomial Operations

• Addition and Subtraction: Combine like terms to add or subtract polynomials. $$(2x^3 + 4x^2 - x + 5) + (x^3 - 3x^2 + 2x - 1) = 3x^3 + x^2 + x + 4$$
• Multiplication: Use distributive property to multiply polynomials. $$(x + 2)(x^2 - x + 3) = x^3 - x^2 + 3x + 2x^2 - 2x + 6 = x^3 + x^2 + 6$$
• Division: Polynomial division can be done using long division or synthetic division.

5. Roots of Polynomials

• The roots of a polynomial are the values of $x$ for which $P(x) = 0$. For example, the roots of $x^2 - 5x + 6 = 0$ are $x = 2$ and $x = 3$.

6. Factor Theorem

• A polynomial $P(x)$ has a factor $(x - a)$ if and only if $P(a) = 0$. This helps in factoring polynomials.

7. Remainder Theorem

• When a polynomial $P(x)$ is divided by $(x - a)$, the remainder is $P(a)$. This theorem is useful for evaluating polynomials.

8. Factorization of Polynomials

• Polynomials can be factored into products of simpler polynomials. For example: $$x^2 - 5x + 6 = (x - 2)(x - 3)$$

9. Polynomial Identities

• Square of a Binomial: $$(a + b)^2 = a^2 + 2ab + b^2$$ $$(a - b)^2 = a^2 - 2ab + b^2$$
• Product of Sum and Difference: $$(a + b)(a - b) = a^2 - b^2$$

Formula Sheet

1. Degree of a Polynomial:

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

2. Addition and Subtraction of Polynomials:

   $$(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0) \pm (b_n x^n + b_{n-1} x^{n-1} + \cdots + b_0)$$

   Combine like terms.

3. Multiplication of Polynomials:

   $$(a_m x^m + a_{m-1} x^{m-1} + \cdots + a_0) \cdot (b_n x^n + b_{n-1} x^{n-1} + \cdots + b_0)$$

   Use distributive property to multiply.

4. Division of Polynomials:

   • Polynomial Long Division: Divide the highest degree terms, subtract, and repeat.
   • Synthetic Division: A shortcut for dividing polynomials when dividing by a linear factor.

5. Factor Theorem:

   $$\text{If } P(a) = 0, \text{ then } (x - a) \text{ is a factor of } P(x)$$

6. Remainder Theorem:

   $$\text{If } P(x) \text{ is divided by } (x - a), \text{ then the remainder is } P(a)$$

7. Factorization of Polynomials:

   • Quadratic Polynomial: $$ax^2 + bx + c = a(x - \alpha)(x - \beta)$$
   • Cubic Polynomial: Factor using known roots or synthetic division.

8. Polynomial Identities:

   • Square of a Binomial: $$(a + b)^2 = a^2 + 2ab + b^2$$ $$(a - b)^2 = a^2 - 2ab + b^2$$
   • Product of Sum and Difference: $$(a + b)(a - b) = a^2 - b^2$$

Examples for Practice

1. Add and simplify the polynomials $3x^2 - 2x + 5$ and $4x^2 + 3x - 2$.
2. Multiply the polynomials $x - 2$ and $x^2 + x + 3$.
3. Use the factor theorem to find the factors of $x^3 - 3x^2 - 4x + 12$.
4. Verify the remainder when dividing $x^3 - 4x^2 + 5x - 6$ by $x - 2$ using the remainder theorem.