Use Synthetic Division To Simplify X 2 + 15 X + 36 X + 1 \frac{x^2+15x+36}{x+1} X + 1 X 2 + 15 X + 36 ​ .Write Your Answer In The Form Q ( X ) + R D ( X ) Q(x) + \frac{r}{d(x)} Q ( X ) + D ( X ) R ​ , Where Q ( X Q(x Q ( X ] Is A Polynomial, R R R Is An Integer, And D ( X D(x D ( X ] Is A Linear Polynomial. Simplify

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Introduction

Synthetic division is a method used to simplify rational expressions by dividing a polynomial by a linear polynomial. This technique is particularly useful when dealing with polynomials of high degree, as it allows us to reduce the degree of the polynomial and simplify the expression. In this article, we will use synthetic division to simplify the rational expression x2+15x+36x+1\frac{x^2+15x+36}{x+1}.

What is Synthetic Division?

Synthetic division is a method of dividing a polynomial by a linear polynomial. It is a shortcut for the long division method and is used to simplify rational expressions. The process involves dividing the polynomial by the linear polynomial, and the result is a quotient and a remainder. The quotient is a polynomial, and the remainder is an integer.

How to Perform Synthetic Division

To perform synthetic division, we need to follow these steps:

  1. Write down the coefficients of the polynomial in a row, with the constant term on the right-hand side.
  2. Write down the root of the linear polynomial (in this case, −1-1) on the left-hand side.
  3. Bring down the first coefficient of the polynomial.
  4. Multiply the root by the first coefficient and write the result below the second coefficient.
  5. Add the second coefficient and the result from step 4.
  6. Multiply the root by the result from step 5 and write the result below the third coefficient.
  7. Add the third coefficient and the result from step 6.
  8. Repeat steps 5-7 until you reach the last coefficient.
  9. The final result is the quotient and the remainder.

Example: Simplifying x2+15x+36x+1\frac{x^2+15x+36}{x+1}

Let's use synthetic division to simplify the rational expression x2+15x+36x+1\frac{x^2+15x+36}{x+1}. We will divide the polynomial x2+15x+36x^2+15x+36 by the linear polynomial x+1x+1.

Step 1: Write down the coefficients of the polynomial

The coefficients of the polynomial x2+15x+36x^2+15x+36 are 1, 15, and 36.

Step 2: Write down the root of the linear polynomial

The root of the linear polynomial x+1x+1 is −1-1.

Step 3: Bring down the first coefficient

The first coefficient is 1.

Step 4: Multiply the root by the first coefficient and write the result below the second coefficient

The root is −1-1, and the first coefficient is 1. Multiplying them together gives −1×1=−1-1 \times 1 = -1. We write this result below the second coefficient, which is 15.

Step 5: Add the second coefficient and the result from step 4

The second coefficient is 15, and the result from step 4 is −1-1. Adding them together gives 15+(−1)=1415 + (-1) = 14.

Step 6: Multiply the root by the result from step 5 and write the result below the third coefficient

The root is −1-1, and the result from step 5 is 14. Multiplying them together gives −1×14=−14-1 \times 14 = -14. We write this result below the third coefficient, which is 36.

Step 7: Add the third coefficient and the result from step 6

The third coefficient is 36, and the result from step 6 is −14-14. Adding them together gives 36+(−14)=2236 + (-14) = 22.

Step 8: The final result is the quotient and the remainder

The final result is the quotient x+14x+14 and the remainder 22.

Conclusion

In this article, we used synthetic division to simplify the rational expression x2+15x+36x+1\frac{x^2+15x+36}{x+1}. We divided the polynomial x2+15x+36x^2+15x+36 by the linear polynomial x+1x+1 and obtained the quotient x+14x+14 and the remainder 22. This is an example of how synthetic division can be used to simplify rational expressions.

Final Answer

The final answer is x+14+22x+1\boxed{x+14 + \frac{22}{x+1}}.

Discussion

Synthetic division is a powerful tool for simplifying rational expressions. It allows us to reduce the degree of the polynomial and simplify the expression. In this example, we used synthetic division to simplify the rational expression x2+15x+36x+1\frac{x^2+15x+36}{x+1}. We obtained the quotient x+14x+14 and the remainder 22. This is an example of how synthetic division can be used to simplify rational expressions.

Related Topics

References

Keywords

  • Synthetic division
  • Rational expressions
  • Polynomial division
  • Long division of polynomials
  • Simplifying rational expressions

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Frequently Asked Questions

Q: What is synthetic division?

A: Synthetic division is a method used to simplify rational expressions by dividing a polynomial by a linear polynomial.

Q: How does synthetic division work?

A: Synthetic division involves dividing the polynomial by the linear polynomial, and the result is a quotient and a remainder. The quotient is a polynomial, and the remainder is an integer.

Q: What are the steps involved in synthetic division?

A: The steps involved in synthetic division are:

  1. Write down the coefficients of the polynomial in a row, with the constant term on the right-hand side.
  2. Write down the root of the linear polynomial (in this case, −1-1) on the left-hand side.
  3. Bring down the first coefficient of the polynomial.
  4. Multiply the root by the first coefficient and write the result below the second coefficient.
  5. Add the second coefficient and the result from step 4.
  6. Multiply the root by the result from step 5 and write the result below the third coefficient.
  7. Add the third coefficient and the result from step 6.
  8. Repeat steps 5-7 until you reach the last coefficient.
  9. The final result is the quotient and the remainder.

Q: What is the quotient and remainder in synthetic division?

A: The quotient is a polynomial, and the remainder is an integer. The quotient is obtained by dividing the polynomial by the linear polynomial, and the remainder is the amount left over after the division.

Q: Can synthetic division be used to simplify any rational expression?

A: No, synthetic division can only be used to simplify rational expressions where the denominator is a linear polynomial.

Q: What are some common applications of synthetic division?

A: Synthetic division is commonly used in algebra and calculus to simplify rational expressions, find the roots of polynomials, and solve systems of equations.

Q: How does synthetic division compare to long division of polynomials?

A: Synthetic division is a shortcut for the long division method and is used to simplify rational expressions. Long division of polynomials is a more general method that can be used to divide polynomials of any degree.

Q: Can synthetic division be used to divide polynomials of any degree?

A: No, synthetic division can only be used to divide polynomials of degree 1 or higher.

Q: What are some common mistakes to avoid when using synthetic division?

A: Some common mistakes to avoid when using synthetic division include:

  • Not writing down the coefficients of the polynomial correctly
  • Not writing down the root of the linear polynomial correctly
  • Not bringing down the first coefficient correctly
  • Not multiplying the root by the first coefficient correctly
  • Not adding the second coefficient and the result from step 4 correctly
  • Not repeating steps 5-7 until you reach the last coefficient

Example Questions

Q: Simplify the rational expression x2+15x+36x+1\frac{x^2+15x+36}{x+1} using synthetic division.

A: To simplify the rational expression x2+15x+36x+1\frac{x^2+15x+36}{x+1} using synthetic division, we need to divide the polynomial x2+15x+36x^2+15x+36 by the linear polynomial x+1x+1. The quotient is x+14x+14 and the remainder is 22.

Q: Simplify the rational expression x3+6x2+11x+6x+1\frac{x^3+6x^2+11x+6}{x+1} using synthetic division.

A: To simplify the rational expression x3+6x2+11x+6x+1\frac{x^3+6x^2+11x+6}{x+1} using synthetic division, we need to divide the polynomial x3+6x2+11x+6x^3+6x^2+11x+6 by the linear polynomial x+1x+1. The quotient is x2+5x+6x^2+5x+6 and the remainder is 0.

Practice Problems

Problem 1: Simplify the rational expression x2+10x+24x+2\frac{x^2+10x+24}{x+2} using synthetic division.

A: To simplify the rational expression x2+10x+24x+2\frac{x^2+10x+24}{x+2} using synthetic division, we need to divide the polynomial x2+10x+24x^2+10x+24 by the linear polynomial x+2x+2. The quotient is x+8x+8 and the remainder is 0.

Problem 2: Simplify the rational expression x3+9x2+24x+16x+1\frac{x^3+9x^2+24x+16}{x+1} using synthetic division.

A: To simplify the rational expression x3+9x2+24x+16x+1\frac{x^3+9x^2+24x+16}{x+1} using synthetic division, we need to divide the polynomial x3+9x2+24x+16x^3+9x^2+24x+16 by the linear polynomial x+1x+1. The quotient is x2+8x+16x^2+8x+16 and the remainder is 0.

Conclusion

Synthetic division is a powerful tool for simplifying rational expressions. It allows us to reduce the degree of the polynomial and simplify the expression. In this article, we used synthetic division to simplify the rational expressions x2+15x+36x+1\frac{x^2+15x+36}{x+1} and x3+6x2+11x+6x+1\frac{x^3+6x^2+11x+6}{x+1}. We obtained the quotients x+14x+14 and x2+5x+6x^2+5x+6 and the remainders 22 and 0, respectively. This is an example of how synthetic division can be used to simplify rational expressions.

Final Answer

The final answer is x+14+22x+1\boxed{x+14 + \frac{22}{x+1}}.

Discussion

Synthetic division is a powerful tool for simplifying rational expressions. It allows us to reduce the degree of the polynomial and simplify the expression. In this article, we used synthetic division to simplify the rational expressions x2+15x+36x+1\frac{x^2+15x+36}{x+1} and x3+6x2+11x+6x+1\frac{x^3+6x^2+11x+6}{x+1}. We obtained the quotients x+14x+14 and x2+5x+6x^2+5x+6 and the remainders 22 and 0, respectively. This is an example of how synthetic division can be used to simplify rational expressions.

Related Topics

References

Keywords

  • Synthetic division
  • Rational expressions
  • Polynomial division
  • Long division of polynomials
  • Simplifying rational expressions