How to Factor Sums and Differences of Perfect Cubes

TL;DR

To factor sums and differences of perfect cubes, use the formulas a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²). Identify a and b by taking the cube roots of the terms, then substitute them into the respective formula to find the factored expression.

Transcript

in this video we're going to focus on factoring sums and difference of cubes so let's say if we want to factor the expression x cubed plus 8. now there is an equation that you want to use and here it is a to the third plus b to the third this is equal to uh a plus b times uh a squared minus a b plus b squared so you need to realize that a to the th... Read More

Key Insights

• 👶 The formula for factoring sums of cubes is a + b (a² - ab + b²), while the formula for factoring differences of cubes is a - b (a² + ab + b²).
• 🥡 Identifying the values of a and b by taking the cube root of the given expressions is crucial.
• ✊ The cube root of a number represents the value that, when raised to the power of 3, gives the original number.
• ❣️ Factors such as x, y, numbers, and their combinations can be used as variables for a and b.

Questions & Answers

Q: How do you factor an expression that involves sums of cubes?

To factor a sum of cubes, identify the values of a and b by taking the cube root of the given expressions. Then plug these values into the formula a + b (a² - ab + b²) to obtain the factored expression.

Q: Can you provide an example of factoring a sum of cubes?

Sure! Let's factor the expression x³ + 8. By taking the cube root of x³, we get a = x. The cube root of 8 is 2, giving us b = 2. Plugging these values into the formula, we have (x + 2)(x² - 2x + 4) as the factored expression.

Q: How do you factor an expression that involves differences of cubes?

To factor a difference of cubes, identify the values of a and b by taking the cube root of the given expressions. Then plug these values into the formula a - b (a² + ab + b²) to obtain the factored expression.

Q: Can you provide an example of factoring a difference of cubes?

Certainly! Let's factor the expression x³ - 125. By taking the cube root of x³, we find a = x. The cube root of 125 is 5, giving us b = 5. Plugging these values into the formula, we have (x - 5)(x² + 5x + 25) as the factored expression.

Summary & Key Takeaways

• The video teaches how to factor expressions that involve sums and differences of cubes using a specific equation.

• To factor a sum of cubes, identify the values of a and b by taking the cube root of the given expressions. Plug these values into the formula to obtain the factored expression.

• The same process applies to factor a difference of cubes, but with a change in the sign between a and b in the formula.