2.2 Polynomials

Polynomials are algebraic expressions consisting of terms in the form \(a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\), where \(a_n, a_{n-1}, \dots, a_0\) are constants, \(x\) is the variable, and \(n\) is a non-negative integer.

Key Concepts:

• Degree of a Polynomial:

  The highest power of the variable in the polynomial. For example, the degree of \(3x^4 + 2x^2 + 1\) is 4.

• Addition and Subtraction:

  Combine like terms to add or subtract polynomials.

• Multiplication:

  Use distributive property or expansion to multiply polynomials.

• Factorization:

  Rewrite a polynomial as a product of its factors using techniques like common factors, grouping, or special identities.

Example 1: Find the degree of the polynomial \(4x^5 + 2x^3 - 7x + 8\).

Solution:

The highest power of \(x\) in the polynomial is 5. Therefore, the degree of the polynomial is 5.

Example 2: Add the polynomials \(2x^3 + 3x^2 - 5\) and \(x^3 - 4x^2 + 7x + 1\).

Solution:

Align and add like terms: \[ (2x^3 + 3x^2 - 5) + (x^3 - 4x^2 + 7x + 1) = (2x^3 + x^3) + (3x^2 - 4x^2) + 7x + (-5 + 1). \] Simplify: \[ 3x^3 - x^2 + 7x - 4. \] Therefore, the sum is \(3x^3 - x^2 + 7x - 4\).

Example 3: Multiply the polynomials \((x+2)\) and \((x^2 - 3x + 4)\).

Solution:

Use the distributive property: \[ (x+2)(x^2 - 3x + 4) = x(x^2 - 3x + 4) + 2(x^2 - 3x + 4). \] Expand: \[ x^3 - 3x^2 + 4x + 2x^2 - 6x + 8. \] Combine like terms: \[ x^3 - x^2 - 2x + 8. \] Therefore, the product is \(x^3 - x^2 - 2x + 8\).

Example 4: Factorize the polynomial \(x^3 + 3x^2 - x - 3\).

Solution:

Group terms and factorize: \[ x^3 + 3x^2 - x - 3 = (x^3 + 3x^2) - (x + 3). \] Factor out common terms: \[ x^2(x + 3) - 1(x + 3). \] Factorize further: \[ (x + 3)(x^2 - 1). \] Use the difference of squares: \[ (x + 3)(x - 1)(x + 1). \] Therefore, the factorized form is \((x + 3)(x - 1)(x + 1)\).