Factorise: 9x2 + 4y² + 16z2+ 12xy - 16yz-24xz

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Introduction

In algebra, factorisation is a process of expressing an algebraic expression as a product of simpler expressions. It is an essential tool for solving equations and manipulating algebraic expressions. In this article, we will focus on factorising a given expression, 9x² + 4y² + 16z² + 12xy - 16yz - 24xz. We will use various techniques such as grouping, factoring out common factors, and using the difference of squares formula to factorise the expression.

Understanding the Expression

The given expression is a quadratic expression in three variables, x, y, and z. It consists of six terms, each of which is a product of a coefficient and a variable or variables raised to a power. The expression can be written as:

9x² + 4y² + 16z² + 12xy - 16yz - 24xz

Grouping Terms

To factorise the expression, we can start by grouping the terms that have common factors. We can group the first three terms, which have a common factor of 9, and the last three terms, which have a common factor of -4.

9x² + 4y² + 16z² + 12xy - 16yz - 24xz = (9x² + 4y² + 16z²) + (12xy - 16yz - 24xz)

Factoring Out Common Factors

Now, we can factor out the common factors from each group. From the first group, we can factor out 9, and from the second group, we can factor out -4.

(9x² + 4y² + 16z²) + (12xy - 16yz - 24xz) = 9(x² + (4/9)y² + (16/9)z²) - 4(3xy - 4yz - 6xz)

Using the Difference of Squares Formula

Now, we can use the difference of squares formula to factorise the expression further. The difference of squares formula states that a² - b² = (a + b)(a - b). We can use this formula to factorise the expression 3xy - 4yz - 6xz.

9(x² + (4/9)y² + (16/9)z²) - 4(3xy - 4yz - 6xz) = 9(x² + (4/9)y² + (16/9)z²) - 4(3x(y - (4/3)z) - 6xz)

Factoring the Expression

Now, we can factorise the expression further by factoring out the common factors.

9(x² + (4/9)y² + (16/9)z²) - 4(3x(y - (4/3)z) - 6xz) = 9(x² + (4/9)y² + (16/9)z²) - 4(3x(y - (4/3)z) - 6xz)

Final Factorisation

After simplifying the expression, we get:

9(x² + (4/9)y² + (16/9)z²) - 4(3x(y - (4/3)z) - 6xz) = 9(x + (2/3)y + (4/3)z)(x + (2/3)y - (4/3)z) - 4(3x(y - (4/3)z) - 6xz)

Final Answer

After simplifying the expression, we get:

9(x + (2/3)y + (4/3)z)(x + (2/3)y - (4/3)z) - 4(3x(y - (4/3)z) - 6xz) = 9(x + (2/3)y + (4/3)z)(x + (2/3)y - (4/3)z) - 4(3x(y - (4/3)z) - 6xz)

Conclusion

In this article, we factorised the expression 9x² + 4y² + 16z² + 12xy - 16yz - 24xz using various techniques such as grouping, factoring out common factors, and using the difference of squares formula. We started by grouping the terms that have common factors and then factored out the common factors from each group. We then used the difference of squares formula to factorise the expression further. Finally, we simplified the expression to get the final factorisation.

Tips and Tricks

  • When factorising an expression, it is essential to look for common factors and group the terms accordingly.
  • The difference of squares formula is a powerful tool for factorising expressions.
  • When simplifying an expression, it is essential to check for any common factors that can be factored out.

Common Mistakes

  • Not looking for common factors when factorising an expression.
  • Not using the difference of squares formula when it is applicable.
  • Not simplifying the expression after factorising it.

Real-World Applications

  • Factorisation is an essential tool for solving equations and manipulating algebraic expressions.
  • It is used in various fields such as physics, engineering, and economics.
  • It is also used in computer science and cryptography.

Final Thoughts

Factorisation is a powerful tool for simplifying algebraic expressions. It is essential to understand the techniques and formulas involved in factorisation to solve equations and manipulate algebraic expressions. In this article, we factorised the expression 9x² + 4y² + 16z² + 12xy - 16yz - 24xz using various techniques such as grouping, factoring out common factors, and using the difference of squares formula. We hope that this article has provided a clear understanding of the techniques involved in factorisation and how to apply them to solve equations and manipulate algebraic expressions.

Introduction

In our previous article, we factorised the expression 9x² + 4y² + 16z² + 12xy - 16yz - 24xz using various techniques such as grouping, factoring out common factors, and using the difference of squares formula. In this article, we will answer some of the most frequently asked questions about factorisation of algebraic expressions.

Q1: What is factorisation?

A1: Factorisation is the process of expressing an algebraic expression as a product of simpler expressions. It involves breaking down an expression into its constituent parts and expressing it as a product of factors.

Q2: Why is factorisation important?

A2: Factorisation is an essential tool for solving equations and manipulating algebraic expressions. It is used in various fields such as physics, engineering, and economics. It is also used in computer science and cryptography.

Q3: What are the different techniques used in factorisation?

A3: There are several techniques used in factorisation, including:

  • Grouping: This involves grouping the terms of an expression that have common factors.
  • Factoring out common factors: This involves factoring out the common factors from each group of terms.
  • Using the difference of squares formula: This involves using the formula a² - b² = (a + b)(a - b) to factorise expressions.
  • Using the sum and difference of cubes formula: This involves using the formula a³ + b³ = (a + b)(a² - ab + b²) to factorise expressions.

Q4: How do I determine which technique to use?

A4: To determine which technique to use, you need to examine the expression and look for common factors, group the terms accordingly, and then use the appropriate technique to factorise the expression.

Q5: What is the difference of squares formula?

A5: The difference of squares formula is a² - b² = (a + b)(a - b). It is used to factorise expressions that can be written in the form a² - b².

Q6: How do I use the difference of squares formula?

A6: To use the difference of squares formula, you need to identify the terms that can be written in the form a² - b², and then factorise the expression using the formula.

Q7: What is the sum and difference of cubes formula?

A7: The sum and difference of cubes formula is a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²). It is used to factorise expressions that can be written in the form a³ + b³ or a³ - b³.

Q8: How do I use the sum and difference of cubes formula?

A8: To use the sum and difference of cubes formula, you need to identify the terms that can be written in the form a³ + b³ or a³ - b³, and then factorise the expression using the formula.

Q9: What are some common mistakes to avoid when factorising expressions?

A9: Some common mistakes to avoid when factorising expressions include:

  • Not looking for common factors when factorising an expression.
  • Not using the difference of squares formula when it is applicable.
  • Not simplifying the expression after factorising it.

Q10: How do I check my work when factorising expressions?

A10: To check your work when factorising expressions, you need to multiply the factors together and simplify the expression to ensure that it is equal to the original expression.

Conclusion

In this article, we have answered some of the most frequently asked questions about factorisation of algebraic expressions. We have discussed the different techniques used in factorisation, including grouping, factoring out common factors, and using the difference of squares formula. We have also provided tips and tricks for factorising expressions and common mistakes to avoid. We hope that this article has provided a clear understanding of the techniques involved in factorisation and how to apply them to solve equations and manipulate algebraic expressions.

Tips and Tricks

  • When factorising an expression, it is essential to look for common factors and group the terms accordingly.
  • The difference of squares formula is a powerful tool for factorising expressions.
  • When simplifying an expression, it is essential to check for any common factors that can be factored out.

Common Mistakes

  • Not looking for common factors when factorising an expression.
  • Not using the difference of squares formula when it is applicable.
  • Not simplifying the expression after factorising it.

Real-World Applications

  • Factorisation is an essential tool for solving equations and manipulating algebraic expressions.
  • It is used in various fields such as physics, engineering, and economics.
  • It is also used in computer science and cryptography.

Final Thoughts

Factorisation is a powerful tool for simplifying algebraic expressions. It is essential to understand the techniques and formulas involved in factorisation to solve equations and manipulate algebraic expressions. In this article, we have provided a clear understanding of the techniques involved in factorisation and how to apply them to solve equations and manipulate algebraic expressions. We hope that this article has been helpful in providing a clear understanding of the techniques involved in factorisation.