Simplify The Expression: ( X + 5 ) ( X 2 + 5 X + 25 (x+5)(x^2+5x+25 ( X + 5 ) ( X 2 + 5 X + 25 ]

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Introduction

In algebra, simplifying expressions is a crucial skill that helps in solving equations and inequalities. It involves combining like terms, removing parentheses, and factoring expressions to make them easier to work with. In this article, we will simplify the given expression $(x+5)(x^2+5x+25)$ using various algebraic techniques.

Understanding the Expression

The given expression is a product of two binomials: $(x+5)$ and $(x^2+5x+25)$. To simplify this expression, we need to multiply each term in the first binomial by each term in the second binomial.

Multiplying Binomials

To multiply binomials, we use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. We will apply this property to each term in the first binomial.

Multiplying the First Term

The first term in the first binomial is $x$. We will multiply this term by each term in the second binomial:

$x(x^2+5x+25) = x^3 + 5x^2 + 25x$

Multiplying the Second Term

The second term in the first binomial is $5$. We will multiply this term by each term in the second binomial:

$5(x^2+5x+25) = 5x^2 + 25x + 125$

Combining Like Terms

Now that we have multiplied each term in the first binomial by each term in the second binomial, we can combine like terms. Like terms are terms that have the same variable raised to the same power.

Combining the $x^2$ Terms

The $x^2$ terms are $5x^2$ and $5x^2$. We can combine these terms by adding their coefficients:

$5x^2 + 5x^2 = 10x^2$

Combining the $x$ Terms

The $x$ terms are $25x$ and $25x$. We can combine these terms by adding their coefficients:

$25x + 25x = 50x$

Combining the Constant Terms

The constant terms are $25$ and $125$. We can combine these terms by adding their coefficients:

$25 + 125 = 150$

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression by writing it in the form $ax^3 + bx^2 + cx + d$.

Writing the Expression in Standard Form

The simplified expression is:

$x^3 + 10x^2 + 50x + 150$

Conclusion

In this article, we simplified the expression $(x+5)(x^2+5x+25)$ using various algebraic techniques. We multiplied binomials, combined like terms, and wrote the expression in standard form. This expression can be used to solve equations and inequalities, and it can also be used to model real-world problems.

Applications of Simplifying Expressions

Simplifying expressions has many applications in mathematics and real-world problems. Some of these applications include:

• Solving Equations and Inequalities: Simplifying expressions is a crucial step in solving equations and inequalities. By simplifying expressions, we can make it easier to isolate the variable and solve for its value.
• Modeling Real-World Problems: Simplifying expressions can be used to model real-world problems. For example, we can use the simplified expression $x^3 + 10x^2 + 50x + 150$ to model the growth of a population or the cost of a product.
• Factoring Expressions: Simplifying expressions can also be used to factor expressions. By simplifying expressions, we can make it easier to factor them and find their roots.

Tips for Simplifying Expressions

Simplifying expressions can be a challenging task, but there are some tips that can make it easier. Some of these tips include:

• Use the Distributive Property: The distributive property is a powerful tool for simplifying expressions. By using the distributive property, we can multiply binomials and combine like terms.
• Combine Like Terms: Combining like terms is a crucial step in simplifying expressions. By combining like terms, we can make it easier to write the expression in standard form.
• Use Algebraic Techniques: Algebraic techniques such as factoring and canceling can be used to simplify expressions. By using these techniques, we can make it easier to simplify expressions and find their roots.

Final Thoughts

Simplifying expressions is a crucial skill in algebra and mathematics. By simplifying expressions, we can make it easier to solve equations and inequalities, model real-world problems, and factor expressions. In this article, we simplified the expression $(x+5)(x^2+5x+25)$ using various algebraic techniques. We hope that this article has provided you with a better understanding of how to simplify expressions and has given you the skills and confidence to tackle more complex problems.

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Introduction

In our previous article, we simplified the expression $(x+5)(x^2+5x+25)$ using various algebraic techniques. In this article, we will answer some of the most frequently asked questions about simplifying expressions.

Q&A

Q: What is the distributive property, and how is it used in simplifying expressions?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. This property is used in simplifying expressions by multiplying each term in the first binomial by each term in the second binomial.

Q: How do I combine like terms in an expression?

A: To combine like terms, you need to identify the terms that have the same variable raised to the same power. Then, you can add or subtract the coefficients of these terms to simplify the expression.

Q: What is the difference between a binomial and a polynomial?

A: A binomial is an algebraic expression that consists of two terms, such as $x+5$ or $x^2+5x+25$. A polynomial, on the other hand, is an algebraic expression that consists of three or more terms, such as $x^3+10x^2+50x+150$.

Q: How do I simplify an expression with multiple binomials?

A: To simplify an expression with multiple binomials, you need to multiply each binomial by each other binomial, and then combine like terms.

Q: What is the purpose of simplifying expressions?

A: The purpose of simplifying expressions is to make them easier to work with and to make it easier to solve equations and inequalities.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

Q: How do I know if an expression is already simplified?

A: An expression is already simplified if it cannot be simplified further by combining like terms or using other algebraic techniques.

Q: Can I simplify an expression with variables in the denominator?

A: Yes, you can simplify an expression with variables in the denominator. However, you need to be careful when simplifying expressions with variables in the denominator, as this can lead to errors.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you need to multiply each term in the numerator and denominator by the least common multiple of the denominators.

Q: Can I simplify an expression with absolute values?

A: Yes, you can simplify an expression with absolute values. However, you need to be careful when simplifying expressions with absolute values, as this can lead to errors.

Conclusion

Simplifying expressions is a crucial skill in algebra and mathematics. By understanding the distributive property, combining like terms, and using other algebraic techniques, you can simplify expressions and make them easier to work with. In this article, we answered some of the most frequently asked questions about simplifying expressions, and we hope that this article has provided you with a better understanding of how to simplify expressions.

Tips for Simplifying Expressions

• Use the Distributive Property: The distributive property is a powerful tool for simplifying expressions. By using the distributive property, you can multiply binomials and combine like terms.
• Combine Like Terms: Combining like terms is a crucial step in simplifying expressions. By combining like terms, you can make it easier to write the expression in standard form.
• Use Algebraic Techniques: Algebraic techniques such as factoring and canceling can be used to simplify expressions. By using these techniques, you can make it easier to simplify expressions and find their roots.
• Check Your Work: It's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

Final Thoughts

Simplifying expressions is a crucial skill in algebra and mathematics. By understanding the distributive property, combining like terms, and using other algebraic techniques, you can simplify expressions and make them easier to work with. In this article, we answered some of the most frequently asked questions about simplifying expressions, and we hope that this article has provided you with a better understanding of how to simplify expressions.