Rewrite The Following Polynomial In Standard Form.\[$-x^4 + 1 - \frac{x}{7}\$\]

Introduction

Polynomials are a fundamental concept in algebra, and understanding how to rewrite them in standard form is crucial for solving equations and manipulating expressions. In this article, we will delve into the process of rewriting the given polynomial in standard form, exploring the steps and techniques involved.

What is a Polynomial in Standard Form?

A polynomial in standard form is a mathematical expression consisting of variables and coefficients, where the variables are raised to non-negative integer powers. The standard form of a polynomial is typically written with the terms arranged in descending order of the powers of the variables. For example, the polynomial $x^2 + 3x - 4$ is in standard form.

Rewriting the Given Polynomial

The given polynomial is $-x^4 + 1 - \frac{x}{7}$. To rewrite this polynomial in standard form, we need to follow a step-by-step approach.

Step 1: Identify the Terms

The given polynomial consists of three terms:

1. $-x^4$
2. $1$
3. $-\frac{x}{7}$

Step 2: Simplify the Terms

We can simplify the terms by combining like terms. However, in this case, the terms are already simplified.

Step 3: Arrange the Terms in Descending Order

To rewrite the polynomial in standard form, we need to arrange the terms in descending order of the powers of the variables. In this case, the term with the highest power is $-x^4$, followed by $-\frac{x}{7}$, and then $1$.

Step 4: Write the Polynomial in Standard Form

The rewritten polynomial in standard form is:

$$-x^4 - \frac{x}{7} + 1$$

Explanation

In the rewritten polynomial, we have combined the like terms and arranged the terms in descending order of the powers of the variables. The term $-x^4$ has the highest power, followed by $-\frac{x}{7}$, and then $1$.

Tips and Tricks

When rewriting polynomials in standard form, it's essential to follow these tips and tricks:

• Identify the terms and simplify them if possible.
• Arrange the terms in descending order of the powers of the variables.
• Use parentheses to group like terms together.
• Combine like terms to simplify the expression.

Conclusion

Rewriting polynomials in standard form is a crucial skill in algebra. By following the steps outlined in this article, you can rewrite the given polynomial in standard form. Remember to identify the terms, simplify them if possible, arrange the terms in descending order of the powers of the variables, and use parentheses to group like terms together. With practice and patience, you'll become proficient in rewriting polynomials in standard form.

Common Mistakes to Avoid

When rewriting polynomials in standard form, it's easy to make mistakes. Here are some common mistakes to avoid:

• Failing to identify like terms.
• Not arranging the terms in descending order of the powers of the variables.
• Not using parentheses to group like terms together.
• Not combining like terms to simplify the expression.

Real-World Applications

Rewriting polynomials in standard form has numerous real-world applications. For example:

• In engineering, polynomials are used to model complex systems and solve equations.
• In economics, polynomials are used to model economic systems and make predictions.
• In computer science, polynomials are used to optimize algorithms and solve problems.

Final Thoughts

Introduction

In our previous article, we explored the process of rewriting polynomials in standard form. However, we know that practice makes perfect, and there's no better way to learn than by asking questions and getting answers. In this article, we'll address some common questions and concerns about rewriting polynomials in standard form.

Q&A

Q: What is the difference between a polynomial and a polynomial in standard form?

A: A polynomial is a mathematical expression consisting of variables and coefficients, where the variables are raised to non-negative integer powers. A polynomial in standard form is a polynomial that is written with the terms arranged in descending order of the powers of the variables.

Q: How do I identify like terms in a polynomial?

A: Like terms are terms that have the same variable raised to the same power. For example, in the polynomial $x^2 + 3x - 4$, the terms $x^2$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I simplify a polynomial?

A: To simplify a polynomial, you need to combine like terms. This involves adding or subtracting the coefficients of like terms. For example, in the polynomial $x^2 + 3x - 4$, you can simplify it by combining the like terms $x^2$ and $3x$ to get $x^2 + 3x$.

Q: How do I arrange the terms in a polynomial in descending order of the powers of the variables?

A: To arrange the terms in a polynomial in descending order of the powers of the variables, you need to start with the term that has the highest power of the variable and then move to the term with the next highest power, and so on. For example, in the polynomial $x^2 + 3x - 4$, the term $x^2$ has the highest power of the variable, so it comes first, followed by the term $3x$, and then the term $-4$.

Q: What is the importance of using parentheses to group like terms together?

A: Using parentheses to group like terms together is essential when rewriting polynomials in standard form. It helps to clarify the expression and make it easier to read and understand. For example, in the polynomial $(x^2 + 3x) - 4$, the parentheses help to group the like terms $x^2$ and $3x$ together.

Q: Can I rewrite a polynomial in standard form if it has a negative exponent?

A: Yes, you can rewrite a polynomial in standard form even if it has a negative exponent. However, you need to be careful when rearranging the terms. For example, in the polynomial $x^{-2} + 3x$, the term $x^{-2}$ has a negative exponent, so you need to rewrite it as $\frac{1}{x^2}$ before rearranging the terms.

Q: How do I rewrite a polynomial in standard form if it has a fraction as a coefficient?

A: To rewrite a polynomial in standard form if it has a fraction as a coefficient, you need to multiply the entire expression by the denominator of the fraction. For example, in the polynomial $\frac{1}{2}x^2 + 3x$, you can rewrite it as $x^2 + 6x$ by multiplying the entire expression by 2.

Conclusion

Rewriting polynomials in standard form is a crucial skill in algebra. By following the steps outlined in this article and addressing common questions and concerns, you can become proficient in rewriting polynomials in standard form. Remember to identify like terms, simplify the expression, arrange the terms in descending order of the powers of the variables, and use parentheses to group like terms together. With practice and patience, you'll become a master of rewriting polynomials in standard form.

Common Mistakes to Avoid

When rewriting polynomials in standard form, it's easy to make mistakes. Here are some common mistakes to avoid:

• Failing to identify like terms.
• Not arranging the terms in descending order of the powers of the variables.
• Not using parentheses to group like terms together.
• Not combining like terms to simplify the expression.
• Not being careful when rearranging terms with negative exponents or fractions as coefficients.

Real-World Applications

Rewriting polynomials in standard form has numerous real-world applications. For example:

• In engineering, polynomials are used to model complex systems and solve equations.
• In economics, polynomials are used to model economic systems and make predictions.
• In computer science, polynomials are used to optimize algorithms and solve problems.

Final Thoughts

Rewriting polynomials in standard form is a fundamental skill in algebra. By following the steps outlined in this article and addressing common questions and concerns, you can become proficient in rewriting polynomials in standard form. Remember to identify like terms, simplify the expression, arrange the terms in descending order of the powers of the variables, and use parentheses to group like terms together. With practice and patience, you'll become a master of rewriting polynomials in standard form.