Polynomials

A polynomial is an algebraic expression consisting of variables (also called indeterminates), coefficients, and non-negative integer exponents, combined using addition, subtraction, and multiplication. Polynomials are fundamental in algebra and have wide applications in mathematics and science.

1. Definition

• A polynomial in variable x is an expression of the form:
  P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0
• Here, a_n, a_{n-1}, ..., a_0 are real numbers (coefficients), and n is a non-negative integer (degree of the polynomial).

2. Types of Polynomials

• Zero Polynomial: All coefficients are zero, e.g., 0.
• Constant Polynomial: Degree 0, e.g., 5.
• Linear Polynomial: Degree 1, e.g., 2x + 3.
• Quadratic Polynomial: Degree 2, e.g., x² + 2x + 1.
• Cubic Polynomial: Degree 3, e.g., x³ - x + 2.

3. Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. For example, the degree of 4x³ + 2x² - 7 is 3.

4. Coefficients, Terms, and Like Terms

• Coefficient: The numerical factor of a term (e.g., in 5x², 5 is the coefficient).
• Term: Each part of a polynomial separated by + or - (e.g., 3x², -2x, 7).
• Like Terms: Terms with the same variable and exponent (e.g., 2x and 5x).

5. Operations on Polynomials

• Addition: Combine like terms.
  Example: (2x + 3) + (x + 5) = 3x + 8
• Subtraction: Subtract like terms.
  Example: (4x² + 2x) - (x² + x) = 3x² + x
• Multiplication: Multiply each term in one polynomial by each term in the other.
  Example: (x + 2)(x + 3) = x² + 5x + 6
• Division: Divide polynomials using long division or synthetic division.

6. Factorization of Polynomials

• Expressing a polynomial as a product of its factors.
• Common methods: taking out common factors, grouping, using identities, splitting the middle term, etc.
• Example: x² + 5x + 6 = (x + 2)(x + 3)

7. Remainder and Factor Theorem

• Remainder Theorem: If a polynomial f(x) is divided by (x - a), the remainder is f(a).
• Factor Theorem: (x - a) is a factor of f(x) if and only if f(a) = 0.

8. Zeros/Roots of a Polynomial

The values of x for which the polynomial becomes zero are called its zeros or roots.
For example, for P(x) = x² - 4, the zeros are x = 2 and x = -2.

9. Applications

• Solving equations
• Modeling real-life situations
• Graphing curves

10. Practice Questions

1. Find the degree and coefficients of the polynomial: 3x⁴ - 2x² + 7.
2. Add: (2x² + 3x + 1) and (x² - x + 4).
3. Factorize: x² + 7x + 10.
4. If f(x) = x² - 5x + 6, find f(2).
5. Is (x - 3) a factor of x² - 9? Justify your answer.

Summary

• Polynomials are algebraic expressions with non-negative integer exponents.
• They can be added, subtracted, multiplied, divided, and factorized.
• Zeros of a polynomial are values that make the polynomial zero.
• Factor and remainder theorems help in finding factors and remainders quickly.