Overview
On this page, we finish our review of factoring methods by investigating important special cases, such as a difference of squares, a perfect square, or a sum or difference of cubes.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Recognize a difference of squares and factor it directly using the formula \(a^2-b^2=(a+b)(a-b)\).
- Recognize a difference of squares when each square is itself a power of a monomial, for example in a difference of 4th powers: \(x^4-y^4=(x^2)^2-(y^2)^2=(x^2+y^2)(x^2-y^2)\)
- Know that a sum of squares cannot be factored (it is a “prime” or “irreducible” polynomial).
- Recognize a perfect square and factor it directly using the formula \(a^2+2ab+b^2=(a+b)^2\).
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Recognize a sum or difference of cubes and factor it directly using the formulas \(a^3+b^3=(a+b)(a^2+ab+b^2)\) or \(a^3-b^3=(a-b)(a^2+ab+b^2)\), respectively, which can be remembered using the mnemonic “SOAP”: “Same sign, Opposite sign, Always Positive”.
- Factor a special product after first factoring out a GCF when necessary.
- Apply all previously learned factoring methods as appropriate to factor many different polynomials.
To prepare for class
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Watch the following videos (by Tyler Wallace) which show how we can factor a difference of squares - while a sum of squares can never be factored:
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Watch the following video (by Tyler Wallace) which shows how we can use the difference of squares formula to factor more general differences, where each term is itself a square of a monomial:
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Watch the following video (by Tyler Wallace) which explains how to recognize and factor a perfect square:
After class
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Watch the following video (by Tyler Wallace) which show how we can factor a sum or difference of cubes:
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Watch the following video (by Tyler Wallace) which reminds us how to first factor out a greatest common factor before applying other methods:
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Watch the following video (by Tyler Wallace) which gives an overall strategy to apply when trying to factor a polynomial:
