Factorize Fully And Simplify Where Possible:1. $x^3 + Y^3$2. $8a^6 + 729$3. ( 3 X + Y ) 3 + ( 3 X − Y ) 3 (3x + Y)^3 + (3x - Y)^3 ( 3 X + Y ) 3 + ( 3 X − Y ) 3

Introduction

Factorizing and simplifying algebraic expressions are essential skills in mathematics, particularly in algebra and calculus. These skills enable us to rewrite complex expressions in a simpler form, making it easier to solve equations and inequalities. In this article, we will factorize and simplify three given expressions: $x^3 + y^3$, $8a^6 + 729$, and $(3x + y)^3 + (3x - y)^3$.

Factorizing $x^3 + y^3$

The expression $x^3 + y^3$ can be factorized using the sum of cubes formula:

$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$

In this case, $a = x$ and $b = y$. Therefore, we can factorize $x^3 + y^3$ as follows:

$$x^3 + y^3 = (x + y)(x^2 - xy + y^2)$$

This is the factorized form of the expression $x^3 + y^3$.

Simplifying $8a^6 + 729$

The expression $8a^6 + 729$ can be simplified by recognizing that $729 = 9^3$. We can rewrite the expression as follows:

$$8a^6 + 729 = 8a^6 + 9^3$$

Using the formula for the sum of cubes, we can rewrite the expression as:

$$8a^6 + 9^3 = (2a^2 + 9)(4a^4 - 18a^2 + 81)$$

However, we can simplify this expression further by recognizing that $8a^6 = (2a^2)^3$. Therefore, we can rewrite the expression as:

$$8a^6 + 729 = (2a^2)^3 + 9^3$$

Using the sum of cubes formula, we can factorize the expression as follows:

$$(2a^2)^3 + 9^3 = (2a^2 + 9)((2a^2)^2 - (2a^2)(9) + 9^2)$$

Simplifying this expression further, we get:

$$(2a^2 + 9)(4a^4 - 18a^2 + 81)$$

This is the simplified form of the expression $8a^6 + 729$.

Factorizing and Simplifying $(3x + y)^3 + (3x - y)^3$

The expression $(3x + y)^3 + (3x - y)^3$ can be factorized using the sum of cubes formula:

$$(a + b)^3 + (a - b)^3 = 2a^3 + 6ab^2$$

In this case, $a = 3x$ and $b = y$. Therefore, we can factorize the expression as follows:

$$(3x + y)^3 + (3x - y)^3 = 2(3x)^3 + 6(3x)(y)^2$$

Simplifying this expression further, we get:

$$(3x + y)^3 + (3x - y)^3 = 2(27x^3) + 6(9x)(y^2)$$

This is the factorized form of the expression $(3x + y)^3 + (3x - y)^3$.

Conclusion

In this article, we factorized and simplified three given expressions: $x^3 + y^3$, $8a^6 + 729$, and $(3x + y)^3 + (3x - y)^3$. We used various algebraic techniques, including the sum of cubes formula, to rewrite the expressions in a simpler form. These skills are essential in mathematics, particularly in algebra and calculus, and can be applied to a wide range of problems.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

• Factorizing: rewriting an expression as a product of simpler expressions.
• Simplifying: rewriting an expression in a simpler form.
• Sum of cubes formula: a formula for factorizing the sum of two cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Introduction

In our previous article, we factorized and simplified three given expressions: $x^3 + y^3$, $8a^6 + 729$, and $(3x + y)^3 + (3x - y)^3$. In this article, we will answer some frequently asked questions (FAQs) related to factorizing and simplifying algebraic expressions.

Q: What is factorizing?

A: Factorizing is the process of rewriting an expression as a product of simpler expressions. This is done by identifying common factors or using algebraic formulas to break down the expression into smaller parts.

Q: What is simplifying?

A: Simplifying is the process of rewriting an expression in a simpler form. This is done by combining like terms, canceling out common factors, or using algebraic formulas to reduce the complexity of the expression.

Q: How do I factorize an expression?

A: To factorize an expression, you need to identify the common factors or use algebraic formulas to break down the expression into smaller parts. Some common factorization techniques include:

• Factoring out a greatest common factor (GCF)
• Using the sum of cubes formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
• Using the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$

Q: How do I simplify an expression?

A: To simplify an expression, you need to combine like terms, cancel out common factors, or use algebraic formulas to reduce the complexity of the expression. Some common simplification techniques include:

• Combining like terms: adding or subtracting terms with the same variable and exponent
• Canceling out common factors: dividing both sides of an equation by a common factor
• Using algebraic formulas: such as the sum of cubes formula or the difference of squares formula

Q: What are some common algebraic formulas?

A: Some common algebraic formulas include:

• Sum of cubes formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
• Difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$
• Sum of squares formula: $a^2 + b^2 = (a + b)^2 - 2ab$
• Difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Q: How do I know when to factorize or simplify an expression?

A: You should factorize or simplify an expression when:

• The expression is complex and difficult to work with
• You need to solve an equation or inequality involving the expression
• You need to find the roots or solutions of the expression

Q: What are some common mistakes to avoid when factorizing or simplifying expressions?

A: Some common mistakes to avoid when factorizing or simplifying expressions include:

• Not identifying all the common factors
• Not using the correct algebraic formulas
• Not combining like terms correctly
• Not canceling out common factors correctly

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to factorizing and simplifying algebraic expressions. We hope that this article has provided you with a better understanding of these important algebraic techniques and has helped you to avoid common mistakes.