2.5 Algebraic Identities

Description

Quick Overview

Algebraic identities are fundamental equations that hold true for any value of their variables.

Standard

This section introduces several important algebraic identities, including the squares of binomials and cubic formulas. It also illustrates their applications in expanding, factoring, and calculating products using the identities.

Detailed

Algebraic Identities

Algebraic identities are equations that remain true regardless of the values of their variables. This section reviews key identities that include:

  • Identity I: \((x + y)^2 = x^2 + 2xy + y^2\)
  • Identity II: \((x - y)^2 = x^2 - 2xy + y^2\)
  • Identity III: \(x^2 - y^2 = (x + y)(x - y)\)
  • Identity IV: \((x + a)(x + b) = x^2 + (a + b)x + ab\)

The section includes examples demonstrating how to find products using these identities and emphasizes their utility for both expansion and factorization. Additionally, more complex identities such as those involving three variables and cubes of binomials are introduced, expanding the toolbox of algebraic techniques.

Example

Factorise:
Let
\( p(x) = x^3 - 25x^2 + 156x - 150. \)

We will also look for all the factors of \(-150\). Some of these are \( \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 25, \pm 30, \pm 50, \pm 75, \pm 150\).

By trial, we find that \( p(1) = 0 \). So, \( 1 \) is a factor of \( p(x) \).

Now we see that \( x^3 - 25x^2 + 156x - 150 = (x - 1)(x^2 - 24x + 150) \) \[ (Why?) \]

We could have also got this by dividing \( p(x) \) by \( (x - 1) \) using the Factor theorem. By splitting the middle term, we have:

$$ \ x^2 - 24x + 150 = (x - 10)(x - 15) $$

So,
\[ p(x) = (x - 1)(x - 10)(x - 15) \]

Similar Question:

Factorise:
Let
\( p(x) = x^3 - 30x^2 + 195x - 270. \)

We will also look for all the factors of \(-270\). Some of these are \( \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 9, \pm 10, \pm 15, \pm 18, \pm 27, \pm 30, \pm 45, \pm 90, \pm 135, \pm 270 \).

By trial, we find that \( p(3) = 0 \). So, \( 3 \) is a factor of \( p(x) \).

Now we see that \( x^3 - 30x^2 + 195x - 270 = (x - 3)(x^2 - 27x + 90) \) \[ (Why?) \]

We could have also got this by dividing \( p(x) \) by \( (x - 3) \) using the Factor theorem. By splitting the middle term, we have:

$$ \ x^2 - 27x + 90 = (x - 9)(x - 10) $$

So,
\[ p(x) = (x - 3)(x - 9)(x - 10) \]

Note: Ensure to verify all factor pairs and confirm through substitution to maintain solution integrity.

Key Concepts

  • Algebraic Identities: Fundamental algebraic equations that hold true for any value.

  • Factorization: The process of writing an expression as a product of its factors.

  • Expansion: Rewriting an expression in an expanded or simplified manner.

Memory Aids

🎵 Rhymes Time

  • When x and y do combine, it's squared, two xy is divine!

📖 Fascinating Stories

  • Imagine two friends x and y; when they hug, they become their own square.

🧠 Other Memory Gems

  • FORESIGHT: Factor, Organize, Rearrange, Expand, Simplify, Highlight Terms.

🎯 Super Acronyms

SCOOPS

  • Squares
  • Cubes
  • Operations
  • Products
  • Simplifications.

Examples

  • Using Identity I: \((x + 3)(x + 3) = (x + 3)^2 = x^2 + 6x + 9\)

  • Factoring: \(49a^2 + 70ab + 25b^2 = (7a + 5b)^2\)

Glossary of Terms

  • Term: Algebraic Identity

    Definition:

    An algebraic equation that remains true for all values of its variables.

  • Term: Expansion

    Definition:

    The process of rewriting an expression in an extended form.

  • Term: Factorization

    Definition:

    The process of breaking down an expression into its components.