Divide $f(x$\] By $d(x$\]. Your Answer Should Be In The Following Format:$\[ \begin{array}{c} \frac{f(x)}{d(x)}=Q(x)+\frac{R(x)}{d(x)} \\ \frac{f(x)}{d(x)}=\frac{12x^3+20x^2+23x+2}{2x+1} \\ R(x)=[?] \end{array} \\]

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Introduction

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Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation in mathematics, with numerous applications in various fields, including engineering, physics, and computer science. In this article, we will explore the process of dividing polynomials, with a focus on the division of two polynomials, $f(x)$ and $d(x)$.

The Division Algorithm

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The division algorithm for polynomials states that any polynomial $f(x)$ can be expressed as the product of a quotient polynomial $Q(x)$ and a divisor polynomial $d(x)$, plus a remainder polynomial $R(x)$. This can be represented mathematically as:

$$\frac{f(x)}{d(x)}=Q(x)+\frac{R(x)}{d(x)}$$

Example: Divide $f(x)$ by $d(x)$

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Let's consider an example to illustrate the division algorithm. Suppose we want to divide the polynomial $f(x)=12x^3+20x^2+23x+2$ by the polynomial $d(x)=2x+1$. We can use the division algorithm to express the result as:

$$\frac{f(x)}{d(x)}=\frac{12x^3+20x^2+23x+2}{2x+1}$$

Finding the Remainder

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To find the remainder $R(x)$, we need to perform the polynomial division. We can do this by dividing the leading term of $f(x)$, which is $12x^3$, by the leading term of $d(x)$, which is $2x$. This gives us the first term of the quotient polynomial, which is $6x^2$. We then multiply $d(x)$ by $6x^2$ and subtract the result from $f(x)$ to obtain a new polynomial.

Performing the Polynomial Division

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To perform the polynomial division, we can use the following steps:

1. Divide the leading term of $f(x)$, which is $12x^3$, by the leading term of $d(x)$, which is $2x$. This gives us the first term of the quotient polynomial, which is $6x^2$.
2. Multiply $d(x)$ by $6x^2$ and subtract the result from $f(x)$ to obtain a new polynomial.
3. Repeat steps 1 and 2 with the new polynomial until we obtain a remainder that is of lower degree than $d(x)$.

Calculating the Quotient and Remainder

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Let's perform the polynomial division using the steps outlined above.

1. Divide the leading term of $f(x)$, which is $12x^3$, by the leading term of $d(x)$, which is $2x$. This gives us the first term of the quotient polynomial, which is $6x^2$.
2. Multiply $d(x)$ by $6x^2$ and subtract the result from $f(x)$ to obtain a new polynomial:

$$\begin{array}{c} (12x^3+20x^2+23x+2)-(12x^3+12x^2) \\ =8x^2+23x+2 \end{array}$$

3. Repeat steps 1 and 2 with the new polynomial until we obtain a remainder that is of lower degree than $d(x)$.

4. Divide the leading term of the new polynomial, which is $8x^2$, by the leading term of $d(x)$, which is $2x$. This gives us the next term of the quotient polynomial, which is $4x$.

5. Multiply $d(x)$ by $4x$ and subtract the result from the new polynomial to obtain another new polynomial:

$$\begin{array}{c} (8x^2+23x+2)-(8x^2+4x) \\ =19x+2 \end{array}$$

3. Repeat steps 1 and 2 with the new polynomial until we obtain a remainder that is of lower degree than $d(x)$.

4. Divide the leading term of the new polynomial, which is $19x$, by the leading term of $d(x)$, which is $2x$. This gives us the next term of the quotient polynomial, which is $\frac{19}{2}$.

5. Multiply $d(x)$ by $\frac{19}{2}$ and subtract the result from the new polynomial to obtain the remainder:

$$\begin{array}{c} (19x+2)-\left(\frac{19}{2}\right)(2x+1) \\ =19x+2-\left(19x+\frac{19}{2}\right) \\ =-\frac{19}{2} \end{array}$$

Conclusion

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In this article, we have explored the process of dividing polynomials, with a focus on the division of two polynomials, $f(x)$ and $d(x)$. We have used the division algorithm to express the result as the product of a quotient polynomial $Q(x)$ and a divisor polynomial $d(x)$, plus a remainder polynomial $R(x)$. We have also performed the polynomial division using the steps outlined above, and obtained the quotient and remainder polynomials.

Final Answer

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The final answer is:

$$\begin{array}{c} \frac{f(x)}{d(x)}=Q(x)+\frac{R(x)}{d(x)} \\ \frac{f(x)}{d(x)}=\frac{12x^3+20x^2+23x+2}{2x+1} \\ R(x)=-\frac{19}{2} \end{array}$$

References

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• [1] "Polynomial Division" by Math Open Reference
• [2] "Polynomial Division" by Wolfram MathWorld
• [3] "Polynomial Division" by Khan Academy
  

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Introduction

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Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. In our previous article, we explored the process of dividing polynomials, with a focus on the division of two polynomials, $f(x)$ and $d(x)$. In this article, we will answer some of the most frequently asked questions about polynomial division.

Q: What is polynomial division?

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A: Polynomial division is the process of dividing one polynomial by another to obtain a quotient and a remainder.

Q: Why do we need to divide polynomials?

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A: Polynomial division is used in a variety of applications, including engineering, physics, and computer science. It is also used to solve equations and to find the roots of polynomials.

Q: How do I perform polynomial division?

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A: To perform polynomial division, you need to follow these steps:

1. Divide the leading term of the dividend polynomial by the leading term of the divisor polynomial.
2. Multiply the divisor polynomial by the result and subtract the product from the dividend polynomial.
3. Repeat steps 1 and 2 until you obtain a remainder that is of lower degree than the divisor polynomial.

Q: What is the remainder in polynomial division?

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A: The remainder in polynomial division is the polynomial that is left over after you have performed the division. It is of lower degree than the divisor polynomial.

Q: How do I find the remainder in polynomial division?

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A: To find the remainder in polynomial division, you need to follow these steps:

1. Perform the polynomial division as described above.
2. The remainder is the polynomial that is left over after you have performed the division.

Q: What is the quotient in polynomial division?

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A: The quotient in polynomial division is the polynomial that is obtained by dividing the dividend polynomial by the divisor polynomial.

Q: How do I find the quotient in polynomial division?

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A: To find the quotient in polynomial division, you need to follow these steps:

1. Perform the polynomial division as described above.
2. The quotient is the polynomial that is obtained by dividing the dividend polynomial by the divisor polynomial.

Q: Can I use polynomial division to solve equations?

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A: Yes, you can use polynomial division to solve equations. By dividing both sides of the equation by the divisor polynomial, you can isolate the variable and solve for its value.

Q: Can I use polynomial division to find the roots of polynomials?

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A: Yes, you can use polynomial division to find the roots of polynomials. By setting the remainder equal to zero and solving for the variable, you can find the roots of the polynomial.

Q: What are some common mistakes to avoid when performing polynomial division?

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A: Some common mistakes to avoid when performing polynomial division include:

• Not following the correct order of operations
• Not multiplying the divisor polynomial by the correct result
• Not subtracting the product from the dividend polynomial
• Not repeating the process until you obtain a remainder that is of lower degree than the divisor polynomial

Conclusion

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In this article, we have answered some of the most frequently asked questions about polynomial division. We have also provided some tips and tricks for performing polynomial division correctly. By following these steps and avoiding common mistakes, you can perform polynomial division with confidence.

Final Answer

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The final answer is:

• Polynomial division is the process of dividing one polynomial by another to obtain a quotient and a remainder.
• The remainder in polynomial division is the polynomial that is left over after you have performed the division.
• The quotient in polynomial division is the polynomial that is obtained by dividing the dividend polynomial by the divisor polynomial.
• Polynomial division can be used to solve equations and to find the roots of polynomials.

References

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• [1] "Polynomial Division" by Math Open Reference
• [2] "Polynomial Division" by Wolfram MathWorld
• [3] "Polynomial Division" by Khan Academy