Lesson Objectives
- Demonstrate an understanding of polynomial long division
- Learn how to divide polynomials using synthetic division
How to Divide Polynomials using Synthetic Division
In our last lesson, we learned how to divide polynomials using long division. In this lesson, we will learn a shortcut called "synthetic division". When we divide a polynomial by a binomial (two-term polynomial) of the form:
x - k
we can use synthetic division. It is important to note that the coefficient of the variable is 1 and the variable is raised to the first power. For emphasis, we can rewrite our binomial as:
1x1 - k
Let's work through a problem step by step. Suppose we saw the following division problem: $$(x^3+6x^2-24x-64) ÷ (x + 8)$$ First and foremost, we need to make sure the dividend is written in standard form. We will create a division bar and place the coefficients of the dividend underneath.
The 1 is the coefficient of x3, the 6 is the coefficient of x2, the -24 is the coefficient of x, and -64 is our constant or coefficient of x0.
When the divisor is in the form of: (x - k), we write the value for k directly to the left of the long division symbol. In our case, we have a divisor of:
(x + 8)
Since this is not in the form of (x - k), we will make a slight change:
(x + 8) = (x - (-8))
Now that it matches our correct form, we can take (-8) and place this to the left of the long division symbol:
We will need to set up three rows. For reference, we will label each row, however, this is not normally done and unnecessary. 
Now that everything is set up, we can begin our synthetic division. We drop the first number inside of the long division symbol (1) down to row 3: 
Now we multiply the number to the left of the long division symbol (-8) by the first entry in row 3 (1). $$-8 \cdot 1=-8$$ The result is written under the 6 in row 1: 
Now we will add the 6 and (-8) and write the result (-2) directly below in row 3: $$6 + (-8)=-2$$ 
We repeat this process until we reach the end. $$-8 \cdot (-2)=16$$ 
$$-24 + 16=-8$$ 
$$-8 \cdot (-8)=64$$ 
$$-64 + 64=0$$ 
Now that we have reached the end, we have to interpret our answer. The rightmost entry in row 3 is the remainder. In this case, a zero indicates that we have no remainder. The other entries starting from the left and working to the right represent the coefficients of the answer in descending order. The answer is a polynomial that is one degree less than the dividend. Since our dividend had a degree of 3 (largest exponent on x was 3), we know our answer has a degree of 2.
Since our largest exponent on the answer is a 2, we start out with the 1 in the leftmost position as the coefficient for x2, the -2 is the coefficient for x1, and -8 is our constant or coefficient for x0. We can write our answer as: $$(x^3+6x^2-24x-64) ÷ (x + 8)=x^2 - 2x - 8$$ Let's look at a few examples.
Example 1: Divide each using synthetic division. $$(4x^4 + 9x^3 + 9x^2 - x - 10) ÷ (x + 1)$$
Answer: $$(4x^4 + 9x^3 + 9x^2 - x - 10) ÷ (x + 1)=4x^3 + 5x^2 + 4x - 5 - \frac{5}{x + 1}$$ Example 2: Divide each using synthetic division. $$(6x^4 - 18x^3 - 6x + 23) ÷ (x - 3)$$ Since we are missing an x2 term, we will rewrite our problem as: $$(6x^4 - 18x^3 + 0x^2 - 6x + 23) ÷ (x - 3)$$ 
Answer: $$(6x^4 - 18x^3 - 6x + 23) ÷ (x - 3)=6x^3 - 6 + \frac{5}{x - 3}$$
x - k
we can use synthetic division. It is important to note that the coefficient of the variable is 1 and the variable is raised to the first power. For emphasis, we can rewrite our binomial as:
1x1 - k
Let's work through a problem step by step. Suppose we saw the following division problem: $$(x^3+6x^2-24x-64) ÷ (x + 8)$$ First and foremost, we need to make sure the dividend is written in standard form. We will create a division bar and place the coefficients of the dividend underneath.
The 1 is the coefficient of x3, the 6 is the coefficient of x2, the -24 is the coefficient of x, and -64 is our constant or coefficient of x0. When the divisor is in the form of: (x - k), we write the value for k directly to the left of the long division symbol. In our case, we have a divisor of:
(x + 8)
Since this is not in the form of (x - k), we will make a slight change:
(x + 8) = (x - (-8))
Now that it matches our correct form, we can take (-8) and place this to the left of the long division symbol:
We will need to set up three rows. For reference, we will label each row, however, this is not normally done and unnecessary. Now that everything is set up, we can begin our synthetic division. We drop the first number inside of the long division symbol (1) down to row 3: Now we multiply the number to the left of the long division symbol (-8) by the first entry in row 3 (1). $$-8 \cdot 1=-8$$ The result is written under the 6 in row 1: Now we will add the 6 and (-8) and write the result (-2) directly below in row 3: $$6 + (-8)=-2$$ We repeat this process until we reach the end. $$-8 \cdot (-2)=16$$ $$-24 + 16=-8$$ $$-8 \cdot (-8)=64$$ $$-64 + 64=0$$ Now that we have reached the end, we have to interpret our answer. The rightmost entry in row 3 is the remainder. In this case, a zero indicates that we have no remainder. The other entries starting from the left and working to the right represent the coefficients of the answer in descending order. The answer is a polynomial that is one degree less than the dividend. Since our dividend had a degree of 3 (largest exponent on x was 3), we know our answer has a degree of 2. Since our largest exponent on the answer is a 2, we start out with the 1 in the leftmost position as the coefficient for x2, the -2 is the coefficient for x1, and -8 is our constant or coefficient for x0. We can write our answer as: $$(x^3+6x^2-24x-64) ÷ (x + 8)=x^2 - 2x - 8$$ Let's look at a few examples.
Example 1: Divide each using synthetic division. $$(4x^4 + 9x^3 + 9x^2 - x - 10) ÷ (x + 1)$$
Answer: $$(4x^4 + 9x^3 + 9x^2 - x - 10) ÷ (x + 1)=4x^3 + 5x^2 + 4x - 5 - \frac{5}{x + 1}$$ Example 2: Divide each using synthetic division. $$(6x^4 - 18x^3 - 6x + 23) ÷ (x - 3)$$ Since we are missing an x2 term, we will rewrite our problem as: $$(6x^4 - 18x^3 + 0x^2 - 6x + 23) ÷ (x - 3)$$ Answer: $$(6x^4 - 18x^3 - 6x + 23) ÷ (x - 3)=6x^3 - 6 + \frac{5}{x - 3}$$ Skills Check:
Example #1
Find each quotient using synthetic division. $$\frac{7x^{3}+ 13x^{2}+ x - 5}{x + 1}$$
Please choose the best answer.
A
$$7x^{2}+ 6x - 6$$
B
$$7x^{2}+ 5x - 3$$
C
$$7x^{2}+ 2x - 1 + \frac{1}{x + 1}$$
D
$$7x^{2}+ 3x - 2 + \frac{3}{x + 1}$$
E
$$7x^2 + 6x - 5$$
Example #2
Find each quotient using synthetic division. $$\frac{x^{3}- 3x^{2}- 16x + 20}{x - 5}$$
Please choose the best answer.
A
$$x^{2}+ 2x - 9 - \frac{8}{x - 5}$$
B
$$x^{2}- 3x + 1 + \frac{1}{x - 5}$$
C
$$x^{2}+ 2x - 6 - \frac{10}{x - 5}$$
D
$$x^{2}+ 4x - 6 - \frac{8}{x - 5}$$
E
$$x^{2}+ 5x - 7 - \frac{7}{x - 5}$$
Example #3
Find each quotient using synthetic division. $$\frac{x^{3}+ 10x^{2}+ 18x - 42}{x + 6}$$
Please choose the best answer.
A
$$x^{2}+ 3x - 8 - \frac{9}{x + 6}$$
B
$$x^{2}+ 3x - 11$$
C
$$x^{2}+ 4x - 6 - \frac{9}{x + 6}$$
D
$$x^{2}+ 4x - 6 - \frac{6}{x + 6}$$
E
$$x^{2}+ 3x - 3 - \frac{7}{x + 6}$$
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