Special Products of Polynomials

Overview

Special products are specific forms of polynomial multiplication that follow unique patterns. Recognizing these patterns can make it easier to multiply polynomials without fully expanding each term. The most common types are:

Types of Special Products

  • Square of a Binomial: \( (a + b)^2 = a^2 + 2ab + b^2 \)
  • Difference of Squares: \( (a + b)(a - b) = a^2 - b^2 \)
  • Cube of a Binomial: \( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \)

Examples

1. Square of a Binomial

Example: \( (x + 5)^2 \)

\( = x^2 + 2(x)(5) + 5^2 \)

\( = x^2 + 10x + 25 \)

2. Difference of Squares

Example: \( (x + 4)(x - 4) \)

\( = x^2 - 4^2 \)

\( = x^2 - 16 \)

3. Cube of a Binomial

Example: \( (x + 2)^3 \)

\( = x^3 + 3(x^2)(2) + 3(x)(2^2) + 2^3 \)

\( = x^3 + 6x^2 + 12x + 8 \)

Practice Questions

  1. Question 1: Square of a Binomial \( (3x + 4)^2 \)
    Solution

    Apply the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):

    \( = (3x)^2 + 2(3x)(4) + 4^2 \)

    \( = 9x^2 + 24x + 16 \)

  2. Question 2: Difference of Squares \( (x + 7)(x - 7) \)
    Solution

    Apply the formula \( (a + b)(a - b) = a^2 - b^2 \):

    \( = x^2 - 7^2 \)

    \( = x^2 - 49 \)

  3. Question 3: Square of a Binomial \( (5x - 3)^2 \)
    Solution

    Apply the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):

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

    \( = 25x^2 - 30x + 9 \)

  4. Question 4: Cube of a Binomial \( (x - 4)^3 \)
    Solution

    Apply the formula \( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \):

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

    \( = x^3 - 12x^2 + 48x - 64 \)

  5. Question 5: Difference of Squares \( (2x + 5)(2x - 5) \)
    Solution

    Apply the formula \( (a + b)(a - b) = a^2 - b^2 \):

    \( = (2x)^2 - 5^2 \)

    \( = 4x^2 - 25 \)

→Dividing Polynomials