Simplify The Expression: F ( X ) = ( 15 X 8 − 8 ) ( 32 + 10 X 4 F(x) = (15x^8 - 8)(32 + 10x^4 F ( X ) = ( 15 X 8 − 8 ) ( 32 + 10 X 4 ]

Introduction

In algebra, simplifying expressions is a crucial skill that helps in solving equations and manipulating mathematical statements. The given expression, $f(x) = (15x^8 - 8)(32 + 10x^4)$, is a product of two polynomials, and simplifying it will result in a more manageable expression. In this article, we will delve into the process of simplifying the given expression using algebraic manipulation techniques.

Understanding the Expression

Before we begin simplifying the expression, let's break it down and understand its components. The given expression is a product of two polynomials:

$$f(x) = (15x^8 - 8)(32 + 10x^4)$$

The first polynomial is $15x^8 - 8$, and the second polynomial is $32 + 10x^4$. To simplify the expression, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Step 1: Apply the Distributive Property

To simplify the expression, we will apply the distributive property to each term in the first polynomial. This means that we will multiply each term in the first polynomial by each term in the second polynomial.

$$f(x) = (15x^8 - 8)(32 + 10x^4)$$

$$f(x) = 15x^8(32 + 10x^4) - 8(32 + 10x^4)$$

Step 2: Multiply Each Term

Now, we will multiply each term in the first polynomial by each term in the second polynomial.

$$f(x) = 15x^8(32 + 10x^4) - 8(32 + 10x^4)$$

$$f(x) = 15x^8(32) + 15x^8(10x^4) - 8(32) - 8(10x^4)$$

Step 3: Simplify Each Term

Now, we will simplify each term by multiplying the coefficients and combining like terms.

$$f(x) = 15x^8(32) + 15x^8(10x^4) - 8(32) - 8(10x^4)$$

$$f(x) = 480x^8 + 150x^{12} - 256 - 80x^4$$

Step 4: Combine Like Terms

Finally, we will combine like terms to simplify the expression.

$$f(x) = 480x^8 + 150x^{12} - 256 - 80x^4$$

$$f(x) = 150x^{12} + 480x^8 - 80x^4 - 256$$

Conclusion

In this article, we simplified the given expression using algebraic manipulation techniques. We applied the distributive property, multiplied each term, and simplified each term to arrive at the final expression. The simplified expression is $150x^{12} + 480x^8 - 80x^4 - 256$. This expression can be used to solve equations and manipulate mathematical statements.

Tips and Tricks

• When simplifying expressions, always apply the distributive property to each term.
• Multiply each term by each term in the second polynomial.
• Simplify each term by multiplying the coefficients and combining like terms.
• Combine like terms to simplify the expression.

Common Mistakes

• Failing to apply the distributive property.
• Not multiplying each term by each term in the second polynomial.
• Not simplifying each term by multiplying the coefficients and combining like terms.
• Not combining like terms to simplify the expression.

Real-World Applications

Simplifying expressions is a crucial skill in algebra that has real-world applications in various fields, including:

• Physics: Simplifying expressions is essential in solving equations that describe physical phenomena, such as motion and energy.
• Engineering: Simplifying expressions is crucial in designing and analyzing complex systems, such as electrical circuits and mechanical systems.
• Computer Science: Simplifying expressions is essential in programming and algorithm design, where complex mathematical expressions are used to solve problems.

Conclusion

Introduction

In our previous article, we simplified the expression $f(x) = (15x^8 - 8)(32 + 10x^4)$ using algebraic manipulation techniques. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

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

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

Q: How do I apply the distributive property to simplify an expression?

A: To apply the distributive property, multiply each term in the first polynomial by each term in the second polynomial. For example, if we have the expression $(a + b)(c + d)$, we would multiply each term in the first polynomial by each term in the second polynomial: $ac + ad + bc + bd$.

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms involves adding or subtracting terms that have the same variable and exponent. Simplifying an expression involves using algebraic manipulation techniques, such as the distributive property, to rewrite the expression in a more manageable form.

Q: How do I know when to combine like terms and when to simplify an expression?

A: Combine like terms when you have a sum or difference of terms with the same variable and exponent. Simplify an expression when you have a product of two or more polynomials and want to rewrite it in a more manageable form.

Q: Can I simplify an expression by combining like terms?

A: Yes, you can simplify an expression by combining like terms. However, this is typically done after applying the distributive property and simplifying the expression using other algebraic manipulation techniques.

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

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

• Failing to apply the distributive property
• Not multiplying each term by each term in the second polynomial
• Not simplifying each term by multiplying the coefficients and combining like terms
• Not combining like terms to simplify the expression

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

A: An expression is already simplified if it cannot be rewritten in a more manageable form using algebraic manipulation techniques. In other words, if you have applied the distributive property and simplified each term, and there are no like terms to combine, then the expression is already simplified.

Q: Can I simplify an expression using a calculator?

A: Yes, you can simplify an expression using a calculator. However, it is generally more beneficial to simplify expressions by hand, as this helps to develop your algebraic skills and understanding of the underlying mathematics.

Conclusion

In conclusion, simplifying expressions is a crucial skill in algebra that has real-world applications in various fields. By applying the distributive property, multiplying each term, simplifying each term, and combining like terms, we can simplify complex expressions and arrive at a more manageable expression. We hope that this Q&A article has provided you with a better understanding of the concepts and techniques involved in simplifying expressions.