Factoring Binomials & Trinomials - Special Cases

Summary

This video covers special cases of factoring, focusing on formulas for factoring perfect squares and cubes. It begins with the difference of perfect squares and walks through examples, then moves on to perfect square trinomials and how to identify and factor them. The video also explains how to factor sums and differences of cubes using specific formulas, providing multiple examples to illustrate the process. Throughout, it emphasizes recognizing patterns in expressions and applying the appropriate factoring techniques.

Takeaways

• 📐 The difference of squares formula is A^2 - B^2 = (A + B)(A - B).
• 🟧 Perfect square trinomials take the form A^2 ± 2AB + B^2 and can be factored as (A ± B)^2.
• 🟢 The sum of cubes is factored as A^3 + B^3 = (A + B)(A^2 - AB + B^2).
• 🔵 The difference of cubes is factored as A^3 - B^3 = (A - B)(A^2 + AB + B^2).
• ✖️ To factor the difference of perfect squares, like x^2 - 16, take the square roots and apply the formula: (x + 4)(x - 4).
• ✅ Recognizing a perfect square trinomial involves verifying that the middle term is twice the product of the square roots of the first and last terms.
• 🟨 Factoring sum of cubes for expressions like x^3 + 8 uses the cube roots and the corresponding sum of cubes formula.
• 🟥 When factoring the difference of cubes for complex expressions like 16x^3 - 54, find the greatest common factor (GCF) first.
• 🔶 GCF simplifies complex cubic expressions, making it easier to apply the difference of cubes formula.
• 🔷 Factorization of expressions with higher powers, such as 125x^3 - 216y^9, involves both cube roots and power manipulations, requiring the use of the standard cube difference formula.

Q & A

What is the formula for factoring a difference of perfect squares?

The formula for factoring a difference of perfect squares is: a² - b² = (a + b)(a - b).

How do you factor a perfect square trinomial in the form a² + 2ab + b²?

A perfect square trinomial in the form a² + 2ab + b² can be factored as (a + b)².

How do you factor a perfect square trinomial in the form a² - 2ab + b²?

A perfect square trinomial in the form a² - 2ab + b² can be factored as (a - b)².

What is the formula for factoring the sum of perfect cubes?

The formula for factoring the sum of perfect cubes is: a³ + b³ = (a + b)(a² - ab + b²).

What is the formula for factoring the difference of perfect cubes?

The formula for factoring the difference of perfect cubes is: a³ - b³ = (a - b)(a² + ab + b²).

How do you factor x² - 16 using the difference of squares formula?

x² - 16 can be factored as (x + 4)(x - 4) using the difference of squares formula, where a = x and b = 4.

How do you determine if a trinomial is a perfect square trinomial?

To determine if a trinomial is a perfect square trinomial, take the square root of the first and last terms, multiply them, and check if the result is half the middle term's coefficient.

How do you factor 9a² + 24a + 16 as a perfect square trinomial?

First, recognize that 9a² is (3a)² and 16 is 4². Since 3a * 4 = 12 and 12 is half of 24, it confirms that it's a perfect square trinomial. The factored form is (3a + 4)².

How do you factor 27x³ + 64 using the sum of cubes formula?

The cube root of 27x³ is 3x and the cube root of 64 is 4. Using the sum of cubes formula, the expression is factored as (3x + 4)(9x² - 12x + 16).

How do you factor 16x³ - 54 using the difference of cubes formula?

First, factor out the greatest common factor, 2, to get 2(8x³ - 27). Using the difference of cubes formula, the expression is factored as 2(2x - 3)(4x² + 6x + 9).