Factoring perfect squares: shared factors | Mathematics II | High School Math | Khan Academy

TL;DR

Two quadratic expressions, 4x^2 + 12x + 9 and 4x^2 - 9, share a common binomial factor of 2x + 3.

Transcript

• [Voiceover] The quadratic expressions 4x squared, plus 12x, plus 9, and 4x squared minus 9, share a common binomial factor. What binomial factor do they share? And I encourage you to pause the video, see if you can figure it out. So, let's do this by taking each of these expressions, and trying to factor them into binomials, and then see if they ... Read More

Key Insights

• 😑 Quadratic expressions can share a common binomial factor.
• 😑 The perfect square polynomial pattern can be used to factor expressions of the form x^2 + bx + c into (x + a)^2.
• 😑 The difference of squares pattern can be used to factor expressions of the form x^2 - y^2 into (x + y)(x - y).

Questions & Answers

Q: How can the first expression, 4x^2 + 12x + 9, be factored?

The expression can be factored into (2x + 3)^2, as it fits the pattern of a perfect square polynomial. The coefficient on the x term is 2 times the product of 2 and 3, which is 2 times 6.

Q: How can the second expression, 4x^2 - 9, be factored?

The expression can be factored into (2x + 3)(2x - 3), as it fits the pattern of a difference of squares. When you have something in the form A^2 - B^2, it is equal to (A + B)(A - B).

Q: What is the common binomial factor of the two expressions?

The common binomial factor is 2x + 3, which appears in both factored forms of the expressions.

Q: What is the significance of finding a common binomial factor in these expressions?

Finding a common binomial factor allows us to simplify and factor the expressions more easily, making it easier to solve equations or manipulate the expressions in algebraic equations.

Summary & Key Takeaways

• The quadratic expressions 4x^2 + 12x + 9 and 4x^2 - 9 share a common binomial factor.

• The first expression can be factored into (2x + 3)^2 using the perfect square polynomial pattern.

• The second expression can be factored into (2x + 3)(2x - 3) using the difference of squares pattern.

• Both expressions have a common binomial factor of 2x + 3.