Section 3.1: Polynomial Basics

This section introduces polynomials, their degree, standard form, and basic operations such as addition, subtraction, and multiplication.

Example 1: Identifying Degree and Terms

Determine the degree and the number of terms of \( P(x) = 4x^5 - 3x^3 + x - 7 \).

Step 1: Highest power of x is 5 → degree = 5

Step 2: Number of terms = 4 (4x^5, -3x^3, x, -7)

Example 2: Adding Polynomials

Add \( P(x) = 3x^3 + 2x^2 - x + 5 \) and \( Q(x) = -x^3 + 4x - 2 \).

Step 1: Combine like terms:

\( (3x^3 - x^3) + (2x^2) + (-x +4x) + (5 - 2) \)

Step 2: Simplify: \( 2x^3 + 2x^2 + 3x + 3 \)

Example 3: Multiplying Polynomials

Multiply \( (x+2)(x^2 - x +3) \).

Step 1: Distribute each term: \( x*(x^2 - x +3) + 2*(x^2 - x +3) \)

Step 2: Simplify: \( x^3 - x^2 +3x +2x^2 -2x +6 = x^3 + x^2 + x +6 \)

Practice Problems

  1. Find the degree and number of terms of \( 5x^4 - 2x^3 + 7x -1 \)
  2. Add \( (2x^3 - x +4) + (x^3 + 3x -5) \)
  3. Subtract \( (3x^2 +2x -1) - (x^2 -x +2) \)
  4. Multiply \( (x+1)(x^2 + 2x +3) \)
  5. Write \( 4x^5 - 3x^3 + x^2 -7 \) in standard form and identify the leading coefficient
← Previous Next →