Factoring Polynomials - Hard Challenge Problems & Special Cases

TL;DR

Learn how to factor perfect square trinomials and use the difference of squares method to factor complex expressions.

Transcript

in this video we're going to focus on factoring difficult problems so let's say if we have this expression a squared plus 6 a b plus 9 b squared minus 49 go ahead and take a minute and factor this particular problem feel free to pause the video now it helps to know this formula a plus b squared is equal to a squared plus two a b plus b squared what... Read More

Key Insights

• 💯 Perfect square trinomials can be factored using a specific formula, a + b^2 = (a + b)^2.
• ❎ The difference of squares method can be used to factor expressions of the form a^2 - b^2, which can be written as (a + b)(a - b).
• ❎ Factoring complex expressions may require combining both the perfect square trinomial and difference of squares methods.
• ❎ Taking the square root of the leading coefficient and the last term can help determine if a trinomial is a perfect square trinomial.
• 🖕 Factoring by splitting the middle term and using grouping is an alternative method, but not used in the examples provided.
• 🤘 Pay attention to signs and distribute negative signs when necessary in the final factored expression.
• 😑 Using the FOIL method to multiply factored expressions can confirm if the factoring was done correctly.

Questions & Answers

Q: How can you determine if a trinomial is a perfect square trinomial?

To check if a trinomial is a perfect square trinomial, take the square root of the leading coefficient and the square root of the last term. If their product is equal to half of the middle term, it is a perfect square trinomial.

Q: What is the difference of squares method for factoring?

The difference of squares method involves factoring expressions of the form a^2 - b^2. It can be factored as (a + b)(a - b).

Q: How do you factor an expression with both a perfect square trinomial and a difference of squares?

First, factor out the perfect square trinomial using the formula a^2 +2ab + b^2 = (a + b)^2. Then, factor the remaining difference of squares expression using the formula a^2 - b^2 = (a + b)(a - b).

Q: Are there any shortcuts or additional methods for factoring complex expressions?

The video also shows a method of factoring by splitting the middle term and using grouping. However, for the given examples, using the perfect square trinomial and difference of squares methods is more straightforward.

Summary & Key Takeaways

• The video explains how to factor perfect square trinomials using the formula a+b^2 = a^2 + 2ab + b^2, and provides an example.

• The video also demonstrates how to factor expressions using the difference of squares method, with step-by-step explanations for two different examples.

• The final example combines both methods to factor a more complex expression.