The Expression $\frac{9 X^2-2}{3 X+1}$ Is Equivalent To:1. $3 X-1-\frac{1}{3 X+1}$2. $3 X-1+\frac{1}{3 X+1}$3. $3 X+1-\frac{1}{3 X+1}$4. $3 X+1+\frac{1}{3 X+1}$

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Introduction

In algebra, simplifying complex expressions is a crucial skill that helps in solving equations and manipulating mathematical statements. One of the techniques used to simplify expressions is polynomial division, which involves dividing a polynomial by another polynomial. In this article, we will explore the simplification of the expression $\frac{9 x^2-2}{3 x+1}$ using polynomial division and identify its equivalent form from the given options.

Understanding the Expression

The given expression is $\frac{9 x^2-2}{3 x+1}$. To simplify this expression, we need to perform polynomial division. The numerator $9 x^2-2$ can be divided by the denominator $3 x+1$ to obtain a quotient and a remainder.

Polynomial Division

To perform polynomial division, we divide the leading term of the numerator ($9 x^2$) by the leading term of the denominator ($3 x$). This gives us $3 x$. We then multiply the entire denominator by $3 x$ and subtract the result from the numerator.

Step 1: Divide the Leading Term

We divide the leading term of the numerator ($9 x^2$) by the leading term of the denominator ($3 x$) to obtain $3 x$.

Step 2: Multiply the Denominator

We multiply the entire denominator ($3 x+1$) by $3 x$ to obtain $9 x^2+3 x$.

Step 3: Subtract the Result

We subtract the result ($9 x^2+3 x$) from the numerator ($9 x^2-2$) to obtain $-3 x-2$.

Step 4: Divide the New Leading Term

We divide the new leading term of the result ($-3 x$) by the leading term of the denominator ($3 x$) to obtain $-1$.

Step 5: Multiply the Denominator

We multiply the entire denominator ($3 x+1$) by $-1$ to obtain $-3 x-1$.

Step 6: Subtract the Result

We subtract the result ($-3 x-1$) from the result ($-3 x-2$) to obtain $-1$.

Simplifying the Expression

After performing polynomial division, we obtain a quotient of $3 x-1$ and a remainder of $-1$. Therefore, the expression $\frac{9 x^2-2}{3 x+1}$ can be simplified as $3 x-1-\frac{1}{3 x+1}$.

Comparison with the Options

We compare the simplified expression $3 x-1-\frac{1}{3 x+1}$ with the given options:

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

The simplified expression matches option 1.

Conclusion

In this article, we simplified the expression $\frac{9 x^2-2}{3 x+1}$ using polynomial division and identified its equivalent form as $3 x-1-\frac{1}{3 x+1}$. This demonstrates the importance of polynomial division in simplifying complex expressions and solving equations in algebra.

Frequently Asked Questions

• What is polynomial division?
• How do you perform polynomial division?
• What is the simplified form of the expression $\frac{9 x^2-2}{3 x+1}$?

Final Answer

The final answer is $\boxed{1}$, which corresponds to the option $3 x-1-\frac{1}{3 x+1}$.

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Introduction

In our previous article, we explored the simplification of the expression $\frac{9 x^2-2}{3 x+1}$ using polynomial division. We identified the equivalent form of the expression as $3 x-1-\frac{1}{3 x+1}$. In this article, we will address some frequently asked questions related to the simplification of the expression and provide additional insights into polynomial division.

Q&A

Q1: What is polynomial division?

A1: Polynomial division is a technique used to divide a polynomial by another polynomial. It involves dividing the leading term of the numerator by the leading term of the denominator to obtain a quotient and a remainder.

Q2: How do you perform polynomial division?

A2: To perform polynomial division, you divide the leading term of the numerator by the leading term of the denominator to obtain a quotient. You then multiply the entire denominator by the quotient and subtract the result from the numerator. You repeat this process until the degree of the remainder is less than the degree of the denominator.

Q3: What is the simplified form of the expression $\frac{9 x^2-2}{3 x+1}$?

A3: The simplified form of the expression $\frac{9 x^2-2}{3 x+1}$ is $3 x-1-\frac{1}{3 x+1}$.

Q4: Why is polynomial division important in algebra?

A4: Polynomial division is important in algebra because it helps in simplifying complex expressions and solving equations. It is a crucial technique used in various mathematical applications, including calculus and engineering.

Q5: Can you provide an example of polynomial division?

A5: Yes, here is an example of polynomial division:

Suppose we want to divide the polynomial $x^2+3x+2$ by the polynomial $x+2$. We can perform polynomial division as follows:

1. Divide the leading term of the numerator ($x^2$) by the leading term of the denominator ($x$) to obtain $x$.
2. Multiply the entire denominator ($x+2$) by $x$ to obtain $x^2+2x$.
3. Subtract the result ($x^2+2x$) from the numerator ($x^2+3x+2$) to obtain $x+2$.
4. Divide the new leading term of the result ($x$) by the leading term of the denominator ($x$) to obtain $1$.
5. Multiply the entire denominator ($x+2$) by $1$ to obtain $x+2$.
6. Subtract the result ($x+2$) from the result ($x+2$) to obtain $0$.

Therefore, the result of the polynomial division is $x+1$ with a remainder of $0$.

Q6: What is the remainder theorem?

A6: The remainder theorem states that if a polynomial $f(x)$ is divided by a linear polynomial $x-a$, then the remainder is equal to $f(a)$.

Q7: Can you provide an example of the remainder theorem?

A7: Yes, here is an example of the remainder theorem:

Suppose we want to find the remainder of the polynomial $f(x)=x^2+3x+2$ when divided by the linear polynomial $x+2$. We can use the remainder theorem as follows:

1. Evaluate the polynomial $f(x)$ at $x=-2$ to obtain $f(-2)=(-2)^2+3(-2)+2=4-6+2=0$.

Therefore, the remainder of the polynomial $f(x)$ when divided by the linear polynomial $x+2$ is $0$.

Conclusion

In this article, we addressed some frequently asked questions related to the simplification of the expression $\frac{9 x^2-2}{3 x+1}$ and provided additional insights into polynomial division. We also discussed the remainder theorem and provided an example of its application. We hope that this article has been helpful in clarifying any doubts you may have had about polynomial division and the remainder theorem.

Final Answer

The final answer is $\boxed{1}$, which corresponds to the option $3 x-1-\frac{1}{3 x+1}$.