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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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Today, let's begin with the square of a sum, which is defined as (a + b)² = a² + 2ab + b². Can anyone tell me what this identity means?
Does it mean we can expand (a + b)² into something simpler?
Exactly! By using this identity, we can simplify calculations rather than multiplying out the binomial. Can anyone give me an example?
If a = 2 and b = 3, then (2 + 3)² should equal 2² + 2(2)(3) + 3²?
Correct! That would give us 25, which matches exactly with the expanded form. Great job!
Next, let’s talk about the difference of squares, defined as a² - b² = (a - b)(a + b). Why do you think this is useful?
It seems like we can easily factor a difference instead of calculating it directly!
Great insight! Can anyone calculate 9² - 4² using this identity?
Sure! 9² - 4² = (9 - 4)(9 + 4) = 5 * 13, which equals 65!
Perfect! You all are grasping these identities really well!
Let's now cover the cubes of sums and differences, specifically, (a + b)³ and (a - b)³. Can anyone write down these identities?
(a + b)³ = a³ + 3a²b + 3ab² + b³ and (a - b)³ = a³ - 3a²b + 3ab² - b³.
Excellent! Does anyone have an example of using one of these identities?
If I use (2 + 1)³, it would be 27. Using the identity gives me 8 + 12 + 3 + 1, which sums to 27.
Great work! This shows how these identities simplify calculations and reveal patterns.
Fundamental equations to simplify and factor polynomial expressions.
An identity that expresses the square of a sum in expanded form.
An identity that gives the expansion for a difference squared.
An identity that allows factoring an expression as a product of its linear factors.
An identity for expanding the cube of a sum.
An identity for expanding the cube of a difference.
When you add and square what do you find? It's a² plus 2ab, plus b² combined!
Imagine a box of candy. If you double the collection, adding will give you more sweets—much like adding is like squaring and distributing.
For difference of squares: Think D is for Difference, just remember as D = (A-B)(A+B).
Remember the acronym PADS for (a + b)² = P = a², A = 2ab, D = b², S = sum of squares.
Term
What is (a - b)²?
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Term
State the difference of squares.
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An equation involving algebraic expressions that holds true for all values of the variables.
An identity that expands (a + b)² to a² + 2ab + b².
An identity that expands (a - b)² to a² - 2ab + b².
An identity stating that a² - b² can be factored as (a - b)(a + b).
An identity that expands (a + b)³ to a³ + 3a²b + 3ab² + b³.
An identity that expands (a - b)³ to a³ - 3a²b + 3ab² - b³.
This section introduces standard algebraic identities essential for simplifying and factoring algebraic expressions.
In this section, students will learn important algebraic identities such as the square and cube of sums and differences, as well as the difference of squares. Understanding these identities aids in the efficient expansion and factorization of algebraic expressions.
Algebraic identities are equations that hold true for all values of the variables involved. They are invaluable tools in algebra, allowing for simpler calculations and proofs.
Square of a sum:
(a + b)² = a² + 2ab + b²
Square of a difference:
(a - b)² = a² - 2ab + b²
Difference of squares:
a² - b² = (a - b)(a + b)
Cube of a sum:
(a + b)³ = a³ + 3a²b + 3ab² + b³
Cube of a difference:
(a - b)³ = a³ - 3a²b + 3ab² - b³
These identities underpin many concepts in algebra and facilitate the factorization and simplification of polynomials, making problem-solving more efficient and accessible.