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What is the Identity a^3+b^3+c^3-3abc?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity a^3+b^3+c^3-3abc is a special algebraic formula that helps simplify complex cubic expressions. It states that a^3+b^3+c^3-3abc is equal to (a+b+c)(a^2+b^2+c^2-ab-bc-ca). This identity is very useful for solving problems quickly.
Simple Example
Quick Example
Imagine you are calculating the volume of three different boxes and then subtracting a combined 'overlap' amount. If the side lengths were a, b, and c, and the overlap was 3abc, this identity helps you find a simpler way to express that total. It's like finding a shortcut to calculate your total marks in three subjects after deducting some penalty points.
Worked Example
Step-by-Step
Let's verify the identity for a=1, b=2, c=3.
Step 1: Calculate the Left Hand Side (LHS) = a^3+b^3+c^3-3abc
LHS = (1)^3 + (2)^3 + (3)^3 - 3(1)(2)(3)
LHS = 1 + 8 + 27 - 18
LHS = 36 - 18
LHS = 18
Step 2: Calculate the Right Hand Side (RHS) = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)
RHS = (1+2+3)((1)^2+(2)^2+(3)^2-(1)(2)-(2)(3)-(3)(1))
RHS = (6)(1+4+9-2-6-3)
RHS = (6)(14-11)
RHS = (6)(3)
RHS = 18
Step 3: Compare LHS and RHS.
Since LHS = 18 and RHS = 18, we see that LHS = RHS.
Answer: The identity holds true for a=1, b=2, c=3, as both sides equal 18.
Why It Matters
This identity is super important in fields like Engineering and Physics for simplifying complex calculations involving shapes and forces. It helps AI/ML engineers understand data patterns and even plays a role in Chemistry for modeling molecular structures. Knowing this identity can open doors to exciting careers in technology and science.
Common Mistakes
MISTAKE: Forgetting the '3abc' term or its sign. Students often write a^3+b^3+c^3 as (a+b+c)(a^2+b^2+c^2-ab-bc-ca). | CORRECTION: Always remember the full identity includes the -3abc term on the LHS.
MISTAKE: Confusing the signs in the second bracket. Students might write (a^2+b^2+c^2+ab+bc+ca). | CORRECTION: The cross-product terms (-ab, -bc, -ca) in the second bracket must all be negative.
MISTAKE: Not knowing the special case: If a+b+c=0, then a^3+b^3+c^3=3abc. Students often forget this shortcut. | CORRECTION: Always check if a+b+c equals zero, as it simplifies the identity greatly.
Practice Questions
Try It Yourself
QUESTION: If a=2, b= -1, c= -1, find the value of a^3+b^3+c^3-3abc using the identity. | ANSWER: 0
QUESTION: Factorize the expression x^3+y^3+27-9xy. | ANSWER: (x+y+3)(x^2+y^2+9-xy-3y-3x)
QUESTION: If a+b+c=6 and a^2+b^2+c^2=14, and ab+bc+ca=11, find the value of a^3+b^3+c^3-3abc. (Hint: First find ab+bc+ca using (a+b+c)^2). | ANSWER: 18
MCQ
Quick Quiz
What is the identity for a^3+b^3+c^3-3abc?
(a+b+c)(a^2+b^2+c^2+ab+bc+ca)
(a+b+c)(a^2+b^2+c^2-ab-bc-ca)
(a+b+c)(a^2+b^2+c^2-3ab-3bc-3ca)
(a-b-c)(a^2+b^2+c^2-ab-bc-ca)
The Correct Answer Is:
B
Option B is the correct form of the identity. Options A, C, and D have incorrect signs or terms in the second bracket, or an incorrect first bracket.
Real World Connection
In the Real World
Engineers designing bridges or buildings use similar algebraic identities to calculate forces and stresses on complex structures, ensuring they are strong and safe. In computer graphics for games or movies, understanding how shapes transform in 3D space often involves such algebraic rules to render realistic movements and objects.
Key Vocabulary
Key Terms
IDENTITY: An equation that is true for all possible values of its variables. | CUBIC EXPRESSION: An algebraic expression where the highest power of the variable is 3. | FACTORIZE: To express a polynomial as a product of two or more polynomials. | ALGEBRAIC FORMULA: A rule or principle expressed in algebraic symbols.
What's Next
What to Learn Next
Great job mastering this identity! Next, you should explore more advanced factorizations of polynomials, especially those involving higher powers. This will help you tackle even more complex problems in algebra and prepare you for competitive exams.
