"Algebraic expressions and identities",

TL;DR

Algebraic expressions combine variables, numbers, and operations, while identities are equations that are always true for any values of their variables. Understanding how to simplify and factor these expressions is key to solving more complex problems in algebra. Mastering common identities can significantly speed up your calculations and problem-solving.

1. The Mental Model

Think of algebraic expressions as recipes where variables are ingredients whose quantities can change, and operations are instructions. Identities are special recipes that always yield the same result, no matter how much of each ingredient you use.

2. The Core Material

Algebra is all about using letters (variables) to represent unknown numbers. An algebraic expression is a combination of these variables, numbers (constants), and mathematical operations (+, -, ×, ÷). For example, 3x + 5 is an expression.

Terms and Coefficients

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In an expression like 4x^2 - 2y + 7, each part separated by a + or - sign is called a term.
* 4x^2 is a term. Here, 4 is the coefficient (the number multiplying the variable part), and x^2 is the variable part.
* -2y is a term. -2 is the coefficient.
* 7 is a term, specifically a constant term because it has no variable.

Types of Expressions

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• Monomial: An expression with one term (e.g., 5x, -7y^2).
• Binomial: An expression with two terms (e.g., x + y, 3a - 4).
• Trinomial: An expression with three terms (e.g., x^2 + 2x - 1).
• Polynomial: A general term for expressions with one or more terms.

Operations on Algebraic Expressions

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You can add, subtract, multiply, and divide expressions. The key is to combine like terms – terms that have the same variables raised to the same powers.

• Addition/Subtraction: (2x + 3y) + (5x - y) = (2x + 5x) + (3y - y) = 7x + 2y
• Multiplication:
  • Monomial by monomial: (3x)(4y) = 12xy
  • Monomial by polynomial: 2x(x + 3) = 2x*x + 2x*3 = 2x^2 + 6x
  • Polynomial by polynomial (use FOIL for binomials): (x + 2)(x + 3) = x*x + x*3 + 2*x + 2*3 = x^2 + 3x + 2x + 6 = x^2 + 5x + 6

Algebraic Identities

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An algebraic identity is an equation that is true for all possible values of the variables involved. They are super useful shortcuts for expanding or factoring expressions.

Here are some common and crucial identities:

1. $(a + b)^2 = a^2 + 2ab + b^2$ (Square of a sum)
2. $(a - b)^2 = a^2 - 2ab + b^2$ (Square of a difference)
3. $(a + b)(a - b) = a^2 - b^2$ (Difference of squares)
4. $(x + a)(x + b) = x^2 + (a + b)x + ab$
5. $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
6. $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$ or a^3 + 3a^2b + 3ab^2 + b^3
7. $(a - b)^3 = a^3 - b^3 - 3ab(a - b)$ or a^3 - 3a^2b + 3ab^2 - b^3
8. $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ (Sum of cubes)
9. $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ (Difference of cubes)

graph TD
    A["Expression or Identity?"] --> B{Does it hold true for ALL values of variables?};
    B -- Yes --> C["Identity (Always True)"];
    B -- No --> D["Expression (Can be simplified)"];
    C --> C1["Useful for quick expansion/factorisation"];
    D --> D1["Can be evaluated for specific values"];
    D1 --> D2{Combine like terms?};
    D2 -- Yes --> D3["Simplify"];
    D2 -- No --> D4["Is it a polynomial?"];
    D4 -- Yes --> D5["Monomial, Binomial, Trinomial"];
    D4 -- No --> D6["Other (e.g., fractional)"];

Factoring Expressions

Factoring is the reverse of expanding. You break down an expression into a product of simpler ones (often binomials). Identities are super helpful here!

• Common Factors: 3x^2 + 6x = 3x(x + 2)
• Using Identities:
  • x^2 - 9 = (x + 3)(x - 3) (using identity 3)
  • y^2 + 6y + 9 = (y + 3)^2 (using identity 1)

3. Worked Example

Let's expand and simplify the expression (2x + 3)^2 - (2x - 3)(2x + 3).

1. Expand the first part using identity 1: $(a + b)^2 = a^2 + 2ab + b^2$
   Here, a = 2x and b = 3.
   $(2x + 3)^2 = (2x)^2 + 2(2x)(3) + (3)^2 = 4x^2 + 12x + 9$

2. Expand the second part using identity 3: $(a - b)(a + b) = a^2 - b^2$
   Here, a = 2x and b = 3.
   $(2x - 3)(2x + 3) = (2x)^2 - (3)^2 = 4x^2 - 9$

3. Substitute these back into the original expression:
   $(4x^2 + 12x + 9) - (4x^2 - 9)$

4. Remove the parentheses (remember to distribute the negative sign):
   $4x^2 + 12x + 9 - 4x^2 + 9$

5. Combine like terms:
   $(4x^2 - 4x^2) + 12x + (9 + 9)$
   $0 + 12x + 18$
   $12x + 18$

So, (2x + 3)^2 - (2x - 3)(2x + 3) simplifies to 12x + 18.

4. Key Takeaways

• Algebraic expressions are combinations of variables, numbers, and operations.
• Terms are parts of an expression separated by + or -, and coefficients are the numerical part of a term.
• You can only add or subtract like terms (same variables, same powers).
• Algebraic identities are equations always true, acting as shortcuts for expanding and factoring.
• Mastering identities like the square of a sum/difference and difference of squares will save you time.
• Factoring is the reverse process of expanding, often using identities in reverse.
• Always be careful with negative signs when distributing them across parentheses.

Common Mistakes to Avoid:
* Forgetting to distribute a negative sign to all terms inside parentheses.
* Incorrectly applying exponents (e.g., (2x)^2 is 4x^2, not 2x^2).
* Confusing (a + b)^2 with a^2 + b^2 (they are not the same!).
* Trying to combine unlike terms (e.g., 3x + 2y cannot be simplified further).

5. Now Try It

Exercise: Simplify the expression: (x - 5)^2 + (x + 1)(x - 1) - x(2x - 7)

What to do:
1. Expand (x - 5)^2 using the square of a difference identity.
2. Expand (x + 1)(x - 1) using the difference of squares identity.
3. Expand x(2x - 7) by distributing x.
4. Combine all the expanded parts, paying close attention to signs.
5. Group and combine any like terms.

What success looks like: Your final simplified expression will be a binomial (two terms) with no parentheses.