MathematicsQuestion 44.1 Factorize Fully And Simplify Where Possible:4.1.1 $x^3 - Y^3$4.1.2 $8x^3 - 27y^3$4.1.3 64 X 6 − 15625 Y 12 64x^6 - 15625y^{12} 64 X 6 − 15625 Y 12

Introduction

In mathematics, factorizing and simplifying algebraic expressions are essential skills that help us solve equations and inequalities. Factorizing involves expressing an expression as a product of simpler expressions, while simplifying involves reducing an expression to its simplest form. In this article, we will focus on factorizing and simplifying three specific algebraic expressions: $x^3 - y^3$, $8x^3 - 27y^3$, and $64x^6 - 15625y^{12}$.

4.1.1 Factorizing and Simplifying $x^3 - y^3$

The expression $x^3 - y^3$ is a difference of cubes. To factorize this expression, we can use the formula:

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

where $a = x$ and $b = y$. Applying this formula, we get:

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

This is the fully factorized form of the expression. We can simplify it further by canceling out any common factors. In this case, there are no common factors, so the expression remains the same.

4.1.2 Factorizing and Simplifying $8x^3 - 27y^3$

The expression $8x^3 - 27y^3$ is also a difference of cubes. To factorize this expression, we can use the same formula as before:

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

where $a = \sqrt{8}x$ and $b = \sqrt{27}y$. Applying this formula, we get:

$8x^3 - 27y^3 = (\sqrt{8}x - \sqrt{27}y)((\sqrt{8}x)^2 + \sqrt{8}x\sqrt{27}y + (\sqrt{27}y)^2)$

Simplifying this expression, we get:

$8x^3 - 27y^3 = 2\sqrt{2}x - 3\sqrt{3}y(4x^2 + 2\sqrt{6}xy + 9y^2)$

This is the fully factorized form of the expression.

4.1.3 Factorizing and Simplifying $64x^6 - 15625y^{12}$

The expression $64x^6 - 15625y^{12}$ is a difference of squares. To factorize this expression, we can use the formula:

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

where $a = 8x^3$ and $b = 125y^6$. Applying this formula, we get:

$64x^6 - 15625y^{12} = (8x^3 - 125y^6)(8x^3 + 125y^6)$

This is the fully factorized form of the expression. We can simplify it further by recognizing that $8x^3 - 125y^6$ is a difference of cubes, which we can factorize using the formula:

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

where $a = 2x$ and $b = 5y^2$. Applying this formula, we get:

$8x^3 - 125y^6 = (2x - 5y^2)((2x)^2 + 2x(5y^2) + (5y^2)^2)$

Simplifying this expression, we get:

$8x^3 - 125y^6 = 2x - 5y^2(4x^2 + 10xy^2 + 25y^4)$

Substituting this expression back into the original expression, we get:

$64x^6 - 15625y^{12} = (2x - 5y^2(4x^2 + 10xy^2 + 25y^4))(8x^3 + 125y^6)$

This is the fully factorized form of the expression.

Conclusion

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Introduction

In our previous article, we factorized and simplified three algebraic expressions: $x^3 - y^3$, $8x^3 - 27y^3$, and $64x^6 - 15625y^{12}$. In this article, we will answer some common questions related to factorizing and simplifying algebraic expressions.

Q: What is the difference of cubes formula?

A: The difference of cubes formula is:

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

where $a$ and $b$ are any real numbers.

Q: How do I apply the difference of cubes formula?

A: To apply the difference of cubes formula, you need to identify the values of $a$ and $b$ in the expression. Then, you can substitute these values into the formula and simplify the expression.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

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

where $a$ and $b$ are any real numbers.

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, you need to identify the values of $a$ and $b$ in the expression. Then, you can substitute these values into the formula and simplify the expression.

Q: Can I factorize an expression that is not a difference of cubes or squares?

A: Yes, you can factorize an expression that is not a difference of cubes or squares. However, you need to use other factorization techniques, such as factoring out common factors or using the distributive property.

Q: How do I simplify an expression after factorizing it?

A: To simplify an expression after factorizing it, you need to cancel out any common factors between the numerator and denominator. You can also combine like terms and simplify the expression further.

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

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

• Not identifying the correct factorization formula to use
• Not simplifying the expression after factorizing it
• Not canceling out common factors between the numerator and denominator
• Not combining like terms

Q: How can I practice factorizing and simplifying expressions?

A: You can practice factorizing and simplifying expressions by working through example problems and exercises. You can also use online resources and practice tests to help you improve your skills.

Conclusion

In this article, we have answered some common questions related to factorizing and simplifying algebraic expressions. We have also provided some tips and techniques for factorizing and simplifying expressions. By practicing factorizing and simplifying expressions, you can improve your skills and become more confident in your ability to solve equations and inequalities involving algebraic expressions.

Additional Resources

• Factorizing and Simplifying Algebraic Expressions: A Guide for Students
• Algebraic Expressions: Factorizing and Simplifying
• Mathway: Factorizing and Simplifying Algebraic Expressions

Practice Problems

1. Factorize and simplify the expression: $x^2 + 5x + 6$
2. Factorize and simplify the expression: $x^3 - 2x^2 - 5x + 6$
3. Factorize and simplify the expression: $x^4 - 4x^3 + 4x^2 - 4x + 1$

Answer Key

1. $(x + 3)(x + 2)$
2. $(x - 1)(x^2 - x - 6)$
3. $(x - 1)^4$