Sum and difference of cubes

Oct 13, 2025 by windailys.com 86 views

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying and factorizing them is an essential skill for students and professionals alike. In this article, we will simplify and factorize three given algebraic expressions, using various techniques such as factoring, canceling, and multiplying.

Expression 1: Simplify and Factorize

Expression 1

${$ \frac{5p^2 + 5p + 5}{p^3 - 1}$}$

To simplify and factorize this expression, we can start by factoring the numerator and denominator separately.

Factoring the Numerator

The numerator can be factored as follows:

${ $5p^2 + 5p + 5 = 5(p^2 + p + 1)\$}$

Factoring the Denominator

The denominator can be factored as follows:

${$ p^3 - 1 = (p - 1)(p^2 + p + 1)$}$

Canceling Common Factors

Now that we have factored the numerator and denominator, we can cancel out the common factors:

${$ \frac{5(p^2 + p + 1)}{(p - 1)(p^2 + p + 1)} = \frac{5}{p - 1}$}$

Therefore, the simplified and factorized form of the first expression is:

${$ \frac{5}{p - 1}$}$

Expression 2: Simplify and Factorize

Expression 2

${$ \frac{x^3 - 27}{9 - x^2} \times \frac{2x^2 + x - 15}{x^2 + 3x^2 + 9}$}$

To simplify and factorize this expression, we can start by factoring the numerator and denominator separately.

Factoring the First Numerator

The first numerator can be factored as follows:

${$ x^3 - 27 = (x - 3)(x^2 + 3x + 9)$}$

Factoring the First Denominator

The first denominator can be factored as follows:

${ $9 - x^2 = (3 - x)(3 + x)\$}$

Factoring the Second Numerator

The second numerator can be factored as follows:

${ $2x^2 + x - 15 = (2x + 5)(x - 3)\$}$

Factoring the Second Denominator

The second denominator can be factored as follows:

${$ x^2 + 3x^2 + 9 = (x + 3)(x^2 + 9)$}$

Canceling Common Factors

Now that we have factored the numerator and denominator, we can cancel out the common factors:

${$ \frac{(x - 3)(x^2 + 3x + 9)}{(3 - x)(3 + x)} \times \frac{(2x + 5)(x - 3)}{(x + 3)(x^2 + 9)} = \frac{(x - 3)^2(2x + 5)}{(3 - x)(3 + x)(x + 3)(x^2 + 9)}$}$

Therefore, the simplified and factorized form of the second expression is:

${$ \frac{(x - 3)^2(2x + 5)}{(3 - x)(3 + x)(x + 3)(x^2 + 9)}$}$

Expression 3: Simplify and Factorize

Expression 3

${$ \frac{9x^2(x3 - 1) - 4(x^3 - 1)}{3x^2 - x - 2} \div \frac{x^2 + 3x^2 + 9}{x^2 + 3x^2 + 9}$}$

To simplify and factorize this expression, we can start by factoring the numerator and denominator separately.

Factoring the Numerator

The numerator can be factored as follows:

${ $9x^2(x^3 - 1) - 4(x^3 - 1) = (x^3 - 1)(9x^2 - 4)\$}$

Factoring the Denominator

The denominator can be factored as follows:

${ $3x^2 - x - 2 = (3x + 2)(x - 1)\$}$

Canceling Common Factors

Now that we have factored the numerator and denominator, we can cancel out the common factors:

${$ \frac{(x^3 - 1)(9x^2 - 4)}{(3x + 2)(x - 1)} \div \frac{x^2 + 3x^2 + 9}{x^2 + 3x^2 + 9} = \frac{(x^3 - 1)(9x^2 - 4)}{(3x + 2)(x - 1)}$}$

Therefore, the simplified and factorized form of the third expression is:

${$ \frac{(x^3 - 1)(9x^2 - 4)}{(3x + 2)(x - 1)}$}$

Conclusion

In this article, we simplified and factorized three given algebraic expressions using various techniques such as factoring, canceling, and multiplying. We started by factoring the numerator and denominator separately, and then canceled out the common factors to obtain the simplified and factorized form of each expression. The simplified and factorized forms of the expressions are:

${$ \frac{5}{p - 1}$}$
${$ \frac{(x - 3)^2(2x + 5)}{(3 - x)(3 + x)(x + 3)(x^2 + 9)}$}$
${$ \frac{(x^3 - 1)(9x^2 - 4)}{(3x + 2)(x - 1)}$}$

Introduction

In our previous article, we simplified and factorized three given algebraic expressions using various techniques such as factoring, canceling, and multiplying. In this article, we will answer some frequently asked questions (FAQs) related to simplifying and factorizing algebraic expressions.

Q: What is the difference between simplifying and factorizing an algebraic expression?

A: Simplifying an algebraic expression involves reducing it to its simplest form by combining like terms, canceling out common factors, and performing other operations. Factorizing an algebraic expression involves expressing it as a product of simpler expressions, such as linear factors or quadratic factors.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, follow these steps:

Combine like terms by adding or subtracting coefficients of the same variable.
Cancel out common factors by dividing both the numerator and denominator by the common factor.
Perform other operations such as multiplying or dividing the expression by a constant.

Q: How do I factor an algebraic expression?

A: To factor an algebraic expression, follow these steps:

Look for common factors in the numerator and denominator.
Factor out the greatest common factor (GCF) from the numerator and denominator.
Use techniques such as grouping, difference of squares, or sum and difference of cubes to factor the expression.

Q: What are some common techniques for factorizing algebraic expressions?

A: Some common techniques for factorizing algebraic expressions include:

Factoring out the GCF: Factor out the greatest common factor from the numerator and denominator.
Grouping: Group the terms in the numerator and denominator into pairs or groups of three.
Difference of squares: Factor the expression as the difference of two squares.
Sum and difference of cubes: Factor the expression as the sum or difference of two cubes.

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

A: You should simplify an algebraic expression when:

The expression is complex and difficult to work with in its current form.
You need to combine like terms or cancel out common factors.
You need to perform other operations such as multiplying or dividing the expression by a constant.

You should factor an algebraic expression when:

The expression can be expressed as a product of simpler expressions.
You need to find the roots or zeros of the expression.
You need to perform operations such as division or multiplication.

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

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

Not combining like terms: Failing to combine like terms can lead to incorrect simplifications.
Not canceling out common factors: Failing to cancel out common factors can lead to incorrect factorizations.
Not using the correct techniques: Using the wrong techniques can lead to incorrect simplifications or factorizations.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to simplifying and factorizing algebraic expressions. We discussed the difference between simplifying and factorizing, how to simplify and factor an algebraic expression, common techniques for factorizing, and common mistakes to avoid. By following these tips and techniques, you can simplify and factor algebraic expressions with confidence.