Find The Quotient Of $12s^5 + 21s^4$ Divided By $3s^2$.Answer: $\square$

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Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another to obtain a quotient and a remainder. In this article, we will focus on finding the quotient of two given algebraic expressions, specifically $12s^5 + 21s^4$ divided by $3s^2$. This process will involve using the rules of polynomial division, including the division algorithm and the concept of leading coefficients.

Understanding Polynomial Division

Polynomial division is a process of dividing a polynomial by another polynomial to obtain a quotient and a remainder. The process involves dividing the leading term of the dividend by the leading term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.

The Division Algorithm

The division algorithm is a fundamental concept in polynomial division that states that any polynomial can be expressed as the product of the divisor and the quotient, plus the remainder. This can be represented mathematically as:

f(x)g(x)=q(x)+r(x)g(x)\frac{f(x)}{g(x)} = q(x) + \frac{r(x)}{g(x)}

where $f(x)$ is the dividend, $g(x)$ is the divisor, $q(x)$ is the quotient, and $r(x)$ is the remainder.

Finding the Quotient

To find the quotient of $12s^5 + 21s^4$ divided by $3s^2$, we will use the division algorithm. We will divide the leading term of the dividend, $12s^5$, by the leading term of the divisor, $3s^2$, to obtain the first term of the quotient.

Step 1: Divide the Leading Term

The leading term of the dividend is $12s^5$, and the leading term of the divisor is $3s^2$. To divide these terms, we will divide the coefficient of the leading term of the dividend by the coefficient of the leading term of the divisor, and then subtract the exponent of the leading term of the divisor from the exponent of the leading term of the dividend.

12s53s2=4s5−2=4s3\frac{12s^5}{3s^2} = 4s^{5-2} = 4s^3

Step 2: Multiply the Divisor by the Result

We will now multiply the entire divisor by the result obtained in Step 1, which is $4s^3$.

3s2×4s3=12s53s^2 \times 4s^3 = 12s^5

Step 3: Subtract the Result from the Dividend

We will now subtract the result obtained in Step 2 from the dividend.

12s5+21s4−12s5=21s412s^5 + 21s^4 - 12s^5 = 21s^4

Step 4: Repeat the Process

We will now repeat the process by dividing the leading term of the result obtained in Step 3, which is $21s^4$, by the leading term of the divisor, $3s^2$.

21s43s2=7s4−2=7s2\frac{21s^4}{3s^2} = 7s^{4-2} = 7s^2

Step 5: Multiply the Divisor by the Result

We will now multiply the entire divisor by the result obtained in Step 4, which is $7s^2$.

3s2×7s2=21s43s^2 \times 7s^2 = 21s^4

Step 6: Subtract the Result from the Dividend

We will now subtract the result obtained in Step 5 from the dividend.

21s4−21s4=021s^4 - 21s^4 = 0

Conclusion

We have now completed the polynomial division process to find the quotient of $12s^5 + 21s^4$ divided by $3s^2$. The quotient is $4s^3 + 7s^2$, and the remainder is $0$. This process demonstrates the use of the division algorithm and the concept of leading coefficients in polynomial division.

Example Problems

  • Find the quotient of $x^3 + 2x^2$ divided by $x^2$.
  • Find the quotient of $3x^4 + 2x^3$ divided by $x^2$.
  • Find the quotient of $x^5 + 2x^4$ divided by $x^3$.

Practice Problems

  • Find the quotient of $2x^4 + 3x^3$ divided by $x^2$.
  • Find the quotient of $x^6 + 2x^5$ divided by $x^4$.
  • Find the quotient of $3x^3 + 2x^2$ divided by $x^2$.

Solutions

  • Find the quotient of $x^3 + 2x^2$ divided by $x^2$: $x + 2$
  • Find the quotient of $3x^4 + 2x^3$ divided by $x^2$: $3x^2 + 2x$
  • Find the quotient of $x^5 + 2x^4$ divided by $x^3$: $x^2 + 2x$

Conclusion

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another to obtain a quotient and a remainder. The process involves using the division algorithm and the concept of leading coefficients. In this article, we have demonstrated the use of polynomial division to find the quotient of $12s^5 + 21s^4$ divided by $3s^2$. We have also provided example problems and practice problems for readers to try on their own.

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another to obtain a quotient and a remainder. In this article, we will answer some frequently asked questions about polynomial division, including the process of dividing polynomials, the use of the division algorithm, and the concept of leading coefficients.

Q: What is polynomial division?

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

Q: How do I divide polynomials?

A: To divide polynomials, you will need to use the division algorithm, which involves dividing the leading term of the dividend by the leading term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend.

Q: What is the division algorithm?

A: The division algorithm is a fundamental concept in polynomial division that states that any polynomial can be expressed as the product of the divisor and the quotient, plus the remainder.

Q: What is the quotient in polynomial division?

A: The quotient in polynomial division is the result of dividing the dividend by the divisor, and it is usually expressed as a polynomial.

Q: What is the remainder in polynomial division?

A: The remainder in polynomial division is the result of subtracting the product of the divisor and the quotient from the dividend, and it is usually expressed as a polynomial.

Q: How do I determine the degree of the quotient and the remainder?

A: To determine the degree of the quotient and the remainder, you will need to look at the degree of the dividend and the divisor. The degree of the quotient will be equal to the degree of the dividend minus the degree of the divisor, and the degree of the remainder will be less than the degree of the divisor.

Q: Can I divide polynomials with different degrees?

A: Yes, you can divide polynomials with different degrees. However, the degree of the quotient will be equal to the degree of the dividend minus the degree of the divisor, and the degree of the remainder will be less than the degree of the divisor.

Q: How do I handle negative coefficients in polynomial division?

A: To handle negative coefficients in polynomial division, you will need to multiply the entire divisor by the negative of the result, and then subtract the product from the dividend.

Q: Can I divide polynomials with complex coefficients?

A: Yes, you can divide polynomials with complex coefficients. However, the process of dividing complex polynomials is more complicated than dividing polynomials with real coefficients.

Q: How do I check my work in polynomial division?

A: To check your work in polynomial division, you will need to multiply the divisor by the quotient and add the remainder to the product. If the result is equal to the dividend, then your work is correct.

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

A: Some common mistakes to avoid in polynomial division include:

  • Not using the division algorithm correctly
  • Not handling negative coefficients correctly
  • Not checking your work
  • Not using the correct degree of the quotient and the remainder

Conclusion

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another to obtain a quotient and a remainder. In this article, we have answered some frequently asked questions about polynomial division, including the process of dividing polynomials, the use of the division algorithm, and the concept of leading coefficients. We hope that this article has been helpful in understanding polynomial division and has provided you with the tools you need to solve problems involving polynomial division.

Example Problems

  • Divide $x^3 + 2x^2$ by $x^2$.
  • Divide $3x^4 + 2x^3$ by $x^2$.
  • Divide $x^5 + 2x^4$ by $x^3$.

Practice Problems

  • Divide $2x^4 + 3x^3$ by $x^2$.
  • Divide $x^6 + 2x^5$ by $x^4$.
  • Divide $3x^3 + 2x^2$ by $x^2$.

Solutions

  • Divide $x^3 + 2x^2$ by $x^2$: $x + 2$
  • Divide $3x^4 + 2x^3$ by $x^2$: $3x^2 + 2x$
  • Divide $x^5 + 2x^4$ by $x^3$: $x^2 + 2x$

Conclusion

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another to obtain a quotient and a remainder. In this article, we have answered some frequently asked questions about polynomial division, including the process of dividing polynomials, the use of the division algorithm, and the concept of leading coefficients. We hope that this article has been helpful in understanding polynomial division and has provided you with the tools you need to solve problems involving polynomial division.