Use Synthetic Division To Find The Quotient And Remainder When $x^3 - 11x + 20$ Is Divided By $x + 4$ By Completing The Parts Below.(a) Complete This Synthetic Division Table.(b) Write Your Answer In The Following Form: Quotient

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a) or (x + a). In this article, we will use synthetic division to find the quotient and remainder when the polynomial $x^3 - 11x + 20$ is divided by $x + 4$.

Step 1: Set Up the Synthetic Division Table

To set up the synthetic division table, we need to write down the coefficients of the polynomial in descending order of powers of x. In this case, the coefficients are 1, 0, -11, and 20.

	1	0	-11	20

Next, we need to write down the value of the divisor, which is -4 in this case.

	1	0	-11	20
-4

Step 2: Bring Down the First Coefficient

The first coefficient is 1, so we bring it down to the next row.

	1	0	-11	20
-4	1

Step 3: Multiply the First Coefficient by the Divisor

We multiply the first coefficient, 1, by the divisor, -4, to get -4.

	1	0	-11	20
-4	1
	-4

Step 4: Add the Product to the Second Coefficient

We add the product, -4, to the second coefficient, 0, to get -4.

	1	0	-11	20
-4	1	-4
	-4

Step 5: Multiply the Result by the Divisor

We multiply the result, -4, by the divisor, -4, to get 16.

	1	0	-11	20
-4	1	-4
	-4	16

Step 6: Add the Product to the Third Coefficient

We add the product, 16, to the third coefficient, -11, to get 5.

	1	0	-11	20
-4	1	-4	5
	-4	16

Step 7: Multiply the Result by the Divisor

We multiply the result, 5, by the divisor, -4, to get -20.

	1	0	-11	20
-4	1	-4	5
	-4	16	-20

Step 8: Add the Product to the Fourth Coefficient

We add the product, -20, to the fourth coefficient, 20, to get 0.

	1	0	-11	20
-4	1	-4	5	0
	-4	16	-20

Conclusion

The final result of the synthetic division is:

Quotient: $x^2 - 4x + 5$ Remainder: 0

Discussion

Synthetic division is a powerful tool for polynomial division. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a) or (x + a). In this article, we used synthetic division to find the quotient and remainder when the polynomial $x^3 - 11x + 20$ is divided by $x + 4$. The final result of the synthetic division is the quotient $x^2 - 4x + 5$ and the remainder 0.

Example

Synthetic division can be used to divide any polynomial by a linear factor. For example, suppose we want to divide the polynomial $x^2 + 5x + 6$ by $x + 2$. We can use synthetic division to find the quotient and remainder.

	1	5	6
-2	1

We bring down the first coefficient, 1.

	1	5	6
-2	1
	-2

We multiply the first coefficient, 1, by the divisor, -2, to get -2.

	1	5	6
-2	1	-2
	-2

We add the product, -2, to the second coefficient, 5, to get 3.

	1	5	6
-2	1	-2	3
	-2

We multiply the result, 3, by the divisor, -2, to get -6.

	1	5	6
-2	1	-2	3
	-2	-6

We add the product, -6, to the third coefficient, 6, to get 0.

	1	5	6
-2	1	-2	3
	-2	-6	0

The final result of the synthetic division is:

Quotient: $x + 3$ Remainder: 0

Conclusion

$$$$

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a) or (x + a). In this article, we will answer some frequently asked questions about synthetic division.

Q: What is synthetic division?

A: Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a) or (x + a).

Q: How do I set up a synthetic division table?

A: To set up a synthetic division table, you need to write down the coefficients of the polynomial in descending order of powers of x. You also need to write down the value of the divisor.

Q: What is the first step in synthetic division?

A: The first step in synthetic division is to bring down the first coefficient.

Q: How do I multiply the first coefficient by the divisor?

A: To multiply the first coefficient by the divisor, you multiply the first coefficient by the value of the divisor.

Q: How do I add the product to the second coefficient?

A: To add the product to the second coefficient, you add the product to the second coefficient.

Q: What is the final result of synthetic division?

A: The final result of synthetic division is the quotient and the remainder.

Q: Can I use synthetic division to divide any polynomial by a linear factor?

A: Yes, you can use synthetic division to divide any polynomial by a linear factor.

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

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

• Not bringing down the first coefficient
• Not multiplying the first coefficient by the divisor
• Not adding the product to the second coefficient
• Not writing down the value of the divisor

Q: How do I check my work when using synthetic division?

A: To check your work when using synthetic division, you can use the following steps:

• Multiply the first coefficient by the divisor
• Add the product to the second coefficient
• Multiply the result by the divisor
• Add the product to the third coefficient
• Continue this process until you reach the last coefficient

Q: Can I use synthetic division to divide a polynomial by a factor of the form (x - a)?

A: Yes, you can use synthetic division to divide a polynomial by a factor of the form (x - a).

Q: Can I use synthetic division to divide a polynomial by a factor of the form (x + a)?

A: Yes, you can use synthetic division to divide a polynomial by a factor of the form (x + a).

Conclusion

Synthetic division is a powerful tool for polynomial division. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a) or (x + a). In this article, we answered some frequently asked questions about synthetic division.

Example Problems

Here are some example problems to help you practice using synthetic division:

• Divide the polynomial $x^2 + 5x + 6$ by $x + 2$
• Divide the polynomial $x^3 - 11x + 20$ by $x + 4$
• Divide the polynomial $x^2 - 3x + 2$ by $x - 1$

Answer Key

Here are the answers to the example problems:

• Divide the polynomial $x^2 + 5x + 6$ by $x + 2$: Quotient = $x + 3$, Remainder = 0
• Divide the polynomial $x^3 - 11x + 20$ by $x + 4$: Quotient = $x^2 - 4x + 5$, Remainder = 0
• Divide the polynomial $x^2 - 3x + 2$ by $x - 1$: Quotient = $x + 1$, Remainder = 0