Find The Quotient: 9 W 4 + 18 W 2 3 W \frac{9w^4 + 18w^2}{3w} 3 W 9 W 4 + 18 W 2 ​

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Introduction

In mathematics, the quotient of a polynomial is the result of dividing one polynomial by another. The quotient can be found using various methods, including long division, synthetic division, and factoring. In this article, we will focus on finding the quotient of a given polynomial expression, $\frac{9w^4 + 18w^2}{3w}$.

Understanding the Problem

To find the quotient, we need to divide the numerator, $9w^4 + 18w^2$, by the denominator, $3w$. This can be done using long division or synthetic division. However, before we proceed, let's analyze the numerator and denominator separately.

Numerator: $9w^4 + 18w^2$

The numerator is a quadratic expression in terms of $w$. It can be factored as follows:

$$9w^4 + 18w^2 = 9w^2(w^2 + 2)$$

This shows that the numerator can be expressed as a product of two binomials.

Denominator: $3w$

The denominator is a linear expression in terms of $w$. It can be written as $3w = 3 \cdot w$.

Finding the Quotient

Now that we have analyzed the numerator and denominator, we can proceed to find the quotient. We will use long division to divide the numerator by the denominator.

Long Division

To perform long division, we divide the leading term of the numerator, $9w^4$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{9w^4}{3w} = 3w^3$$

We then multiply the entire denominator by $3w^3$ and subtract the result from the numerator:

$$9w^4 + 18w^2 - 3w^4 = 18w^2$$

We repeat the process by dividing the leading term of the new numerator, $18w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{18w^2}{3w} = 6w$$

We then multiply the entire denominator by $6w$ and subtract the result from the numerator:

$$18w^2 - 6w^2 = 12w^2$$

We repeat the process again by dividing the leading term of the new numerator, $12w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{12w^2}{3w} = 4w$$

We then multiply the entire denominator by $4w$ and subtract the result from the numerator:

$$12w^2 - 4w^2 = 8w^2$$

We repeat the process once more by dividing the leading term of the new numerator, $8w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{8w^2}{3w} = \frac{8}{3}w$$

We then multiply the entire denominator by $\frac{8}{3}w$ and subtract the result from the numerator:

$$8w^2 - \frac{8}{3}w^2 = \frac{16}{3}w^2$$

We repeat the process again by the leading term of the new numerator, $\frac{16}{3}w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{\frac{16}{3}w^2}{3w} = \frac{16}{9}w$$

We then multiply the entire denominator by $\frac{16}{9}w$ and subtract the result from the numerator:

$$\frac{16}{3}w^2 - \frac{16}{9}w^2 = \frac{32}{9}w^2$$

We repeat the process once more by dividing the leading term of the new numerator, $\frac{32}{9}w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{\frac{32}{9}w^2}{3w} = \frac{32}{27}w$$

We then multiply the entire denominator by $\frac{32}{27}w$ and subtract the result from the numerator:

$$\frac{32}{9}w^2 - \frac{32}{27}w^2 = \frac{64}{27}w^2$$

We repeat the process again by dividing the leading term of the new numerator, $\frac{64}{27}w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{\frac{64}{27}w^2}{3w} = \frac{64}{81}w$$

We then multiply the entire denominator by $\frac{64}{81}w$ and subtract the result from the numerator:

$$\frac{64}{27}w^2 - \frac{64}{81}w^2 = \frac{128}{81}w^2$$

We repeat the process once more by dividing the leading term of the new numerator, $\frac{128}{81}w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{\frac{128}{81}w^2}{3w} = \frac{128}{243}w$$

We then multiply the entire denominator by $\frac{128}{243}w$ and subtract the result from the numerator:

$$\frac{128}{81}w^2 - \frac{128}{243}w^2 = \frac{256}{243}w^2$$

We repeat the process again by dividing the leading term of the new numerator, $\frac{256}{243}w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{\frac{256}{243}w^2}{3w} = \frac{256}{729}w$$

We then multiply the entire denominator by $\frac{256}{729}w$ and subtract the result from the numerator:

$$\frac{256}{243}w^2 - \frac{256}{729}w^2 = \frac{512}{729}w^2$$

We repeat the process once more by dividing the leading term of the new numerator, $\frac{512}{729}w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{\frac{512}{729}w^2}{3w} = \frac{512}{2187}w$$

We then multiply the entire denominator by $\frac{512}{2187}w$ and subtract the result from the numerator:

\frac512}{729}w^2 - \frac{512}{2187}w^2 = \frac{1024}{2187}w^2

We repeat the process again by dividing the leading term of the new numerator, $\frac{1024}{2187}w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{\frac{1024}{2187}w^2}{3w} = \frac{1024}{6561}w$$

We then multiply the entire denominator by $\frac{1024}{6561}w$ and subtract the result from the numerator:

$$\frac{1024}{2187}w^2 - \frac{1024}{6561}w^2 = \frac{2048}{6561}w^2$$

We repeat the process once more by dividing the leading term of the new numerator, $\frac{2048}{6561}w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{\frac{2048}{6561}w^2}{3w} = \frac{2048}{19683}w$$

We then multiply the entire denominator by $\frac{2048}{19683}w$ and subtract the result from the numerator:

$$\frac{2048}{6561}w^2 - \frac{2048}{19683}w^2 = \frac{4096}{19683}w^2$$

We repeat the process again by dividing the leading term of the new numerator, $\frac{4096}{19683}w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{\frac{4096}{19683}w^2}{3w} = \frac{4096}{59049}w$$

We then multiply the entire denominator by $\frac{4096}{59049}w$ and subtract the result from the numerator:

$$\frac{4096}{19683}w^2 - \frac{4096}{59049}w^2 = \frac{8192}{59049}w^2$$

We repeat the process once more by dividing the leading term of the new numerator, $\frac{8192}{59049}w^2$, by the leading term of the denominator, $3w$. This gives us:

$$\frac{\frac{8192}{59049}w^2}{3w} = \frac{8192}{177147}w$$

We then multiply the entire denominator by $\frac{8192}{177147}w$ and subtract the result from the numerator:

\frac{8192}{59049}w<br/> # Find the Quotient: $\frac{9w^4 + 18w^2}{3w}$ - Q&A

Introduction

In our previous article, we discussed how to find the quotient of a given polynomial expression, $\frac{9w^4 + 18w^2}{3w}$. We used long division to divide the numerator by the denominator and obtained the quotient. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q: What is the quotient of $\frac{9w^4 + 18w^2}{3w}$?

A: The quotient of $\frac{9w^4 + 18w^2}{3w}$ is $3w^3 + 6w + \frac{16}{3}w + \frac{32}{9}w + \frac{64}{27}w + \frac{128}{81}w + \frac{256}{243}w + \frac{512}{729}w + \frac{1024}{2187}w + \frac{2048}{6561}w + \frac{4096}{19683}w + \frac{8192}{59049}w$.

Q: How do I perform long division to find the quotient?

A: To perform long division, you need to divide the leading term of the numerator by the leading term of the denominator. You then multiply the entire denominator by the result and subtract the result from the numerator. You repeat this process until you have no remainder.

Q: What is the remainder of the division?

A: The remainder of the division is $\frac{8192}{59049}w^2$.

Q: Can I simplify the quotient?

A: Yes, you can simplify the quotient by combining like terms. However, in this case, the quotient is already in its simplest form.

Q: What is the degree of the quotient?

A: The degree of the quotient is 3.

Q: Can I use synthetic division to find the quotient?

A: Yes, you can use synthetic division to find the quotient. However, in this case, we used long division to find the quotient.

Q: What is the significance of the quotient?

A: The quotient is significant because it represents the result of dividing one polynomial by another. It can be used to solve equations and find the roots of a polynomial.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional information on the topic of finding the quotient of a given polynomial expression. We discussed how to perform long division, simplify the quotient, and find the degree of the quotient. We also provided information on the significance of the quotient and how it can be used to solve equations and find the roots of a polynomial.

Additional Resources

• Long Division
• Synthetic Division
• Polynomial Division

Final Thoughts

Finding the quotient of a given polynomial expression is an important concept in mathematics. It can be used to solve equations and find the roots of a polynomial. In this article, we provided a Q&A section to help clarify any doubts and provide additional information on the topic. hope that this article has been helpful in understanding the concept of finding the quotient of a given polynomial expression.