What is (ax + by + cz)² Expansion? | Simple Explanation for Class 6

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What is the Factorisation of (ax + by + cz)²?

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Factorisation is breaking down a number or an expression into its basic building blocks (factors) that multiply together to give the original number or expression. For (ax + by + cz)², it means expanding the expression by multiplying it by itself.

Simple Example

Quick Example

Imagine you have a square cricket field with sides of length 'a' + 'b' + 'c' units. To find the total area, you would multiply (a + b + c) by itself. This is similar to expanding (ax + by + cz)², where you multiply the expression by itself to find all the parts that make up the total.

Worked Example

Step-by-Step

Let's expand (2x + 3y + 4z)².
---Step 1: Understand that (2x + 3y + 4z)² means (2x + 3y + 4z) multiplied by (2x + 3y + 4z).
---Step 2: Use the algebraic identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. Here, a = 2x, b = 3y, and c = 4z.
---Step 3: Substitute these values into the identity.
(2x)² + (3y)² + (4z)² + 2(2x)(3y) + 2(3y)(4z) + 2(4z)(2x)
---Step 4: Calculate each term.
(2x)² = 4x²
(3y)² = 9y²
(4z)² = 16z²
2(2x)(3y) = 12xy
2(3y)(4z) = 24yz
2(4z)(2x) = 16zx
---Step 5: Combine all the terms.
4x² + 9y² + 16z² + 12xy + 24yz + 16zx
---Answer: The factorisation (expansion) of (2x + 3y + 4z)² is 4x² + 9y² + 16z² + 12xy + 24yz + 16zx.

Why It Matters

Understanding how to expand expressions like this is crucial in many fields. Engineers use it to design structures, while computer scientists use similar logic in algorithms. Even in data science, these expansions help in creating models to predict trends, like mobile data usage or stock prices.

Common Mistakes

MISTAKE: Only squaring the individual terms, like writing (ax + by + cz)² as a²x² + b²y² + c²z². | CORRECTION: Remember that the square of a sum also includes the 'cross-product' terms (2ab, 2bc, 2ca). The correct expansion is a²x² + b²y² + c²z² + 2axby + 2bycz + 2czax.

MISTAKE: Forgetting the '2' in the cross-product terms, e.g., writing axby instead of 2axby. | CORRECTION: The identity (a+b+c)² always has '2' multiplied with each pair of terms (2ab, 2bc, 2ca). Don't forget this factor of 2!

MISTAKE: Confusing (ax + by + cz)² with a²x² + b²y² + c²z². | CORRECTION: (ax + by + cz)² means (ax + by + cz) multiplied by itself, which generates many more terms than just squaring each part separately. Always use the identity (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca.

Practice Questions

Try It Yourself

QUESTION: Expand (p + 2q + 3r)² | ANSWER: p² + 4q² + 9r² + 4pq + 12qr + 6rp

QUESTION: Find the expansion of (x - y + z)². (Hint: Treat -y as +(-y)) | ANSWER: x² + y² + z² - 2xy - 2yz + 2zx

QUESTION: If (a + b + c)² = 100 and a² + b² + c² = 40, what is the value of (ab + bc + ca)? | ANSWER: 30

MCQ

Quick Quiz

Which of the following is the correct expansion of (m + 2n + 5p)²?

m² + 4n² + 25p²

m² + 4n² + 25p² + 4mn + 10np + 20pm

m² + 4n² + 25p² + 4mn + 20np + 10pm

m² + 2n² + 5p² + 2mn + 10np + 5pm

The Correct Answer Is:

Option C is correct because it correctly applies the identity (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca. Option A only squares individual terms. Options B and D have errors in the cross-product terms.

Real World Connection

In the Real World

This concept is used when calculating areas of complex shapes in architecture or engineering, like designing a new stadium. It also helps in computer graphics to transform and scale objects, or in economics to model how different factors (like price, demand, and supply) interact to determine market outcomes.

Key Vocabulary

Key Terms

Factorisation: Breaking down an expression into simpler parts that multiply together | Expansion: The process of multiplying out brackets in an algebraic expression | Identity: An equation that is true for all possible values of its variables | Term: A single number or variable, or numbers and variables multiplied together | Coefficient: A number multiplied by a variable in an algebraic expression

What's Next

What to Learn Next

Great job understanding expansion! Next, you should explore factorisation of quadratic expressions (like ax² + bx + c). This will help you reverse the process you just learned and find the original factors, which is super useful in solving equations!