Lesson Objectives
- Demonstrate an understanding of polynomial long division
- Learn how to divide polynomials when a term is missing from the dividend
- Learn how to divide polynomials when a term is missing from the divisor
How to Divide Polynomials with Missing Terms
In our last lesson, we learned how to divide polynomials using long division. In some cases, we will have to divide polynomials with missing terms. A polynomial has missing terms when one or more of its powers of the given variable are missing. Let's suppose we had the following:
x4 + x3 + x + 5
Our polynomial above has a missing x2 term. We can see we have x to the 4th power, 3rd power, and 1st power. We are missing x to the 2nd power. When we perform division with polynomials and a term is missing, we use 0 as a coefficient for each missing term. We could write our example polynomial as:
x4 + x3 + 0x2 + x + 5
This is legal since 0 times any number results in 0:
0 • x2 = 0
When we add 0 to a term, the term stays unchanged. Therefore, the 0x2 does not add any value to the polynomial, it is just used as a placeholder for the missing term of x2. Let's take a look at a few examples.
Example 1: Find each quotient.
(x3 + 100 - 14x2) ÷ (-5 + x)
First and foremost, we want to write the dividend and divisor in standard form:
(x3 - 14x2 + 100) ÷ (x - 5)
We can see we are missing an x to the first power term in our dividend. We can write in 0x as a placeholder:
(x3 - 14x2 + 0x + 100) ÷ (x - 5)
Now we can perform our long division:
We can state our answer as: $$x^2 - 9x - 45 + \frac{-125}{x - 5}$$ The same process is used when our divisor is missing a term. Let's look at an example.
Example 2: Find each quotient.
(x4 + 3x3 - 3x2 + 12x - 28) ÷ (x2 + 4)
Each polynomial is already in standard form. Our divisor is missing a term of x to the first power. We can write in 0x as a placeholder:
(x4 + 3x3 - 3x2 + 12x - 28) ÷ (x2 + 0x + 4)
Now we can perform our long division:
We can state our answer as: $$x^2 + 3x - 7$$
x4 + x3 + x + 5
Our polynomial above has a missing x2 term. We can see we have x to the 4th power, 3rd power, and 1st power. We are missing x to the 2nd power. When we perform division with polynomials and a term is missing, we use 0 as a coefficient for each missing term. We could write our example polynomial as:
x4 + x3 + 0x2 + x + 5
This is legal since 0 times any number results in 0:
0 • x2 = 0
When we add 0 to a term, the term stays unchanged. Therefore, the 0x2 does not add any value to the polynomial, it is just used as a placeholder for the missing term of x2. Let's take a look at a few examples.
Example 1: Find each quotient.
(x3 + 100 - 14x2) ÷ (-5 + x)
First and foremost, we want to write the dividend and divisor in standard form:
(x3 - 14x2 + 100) ÷ (x - 5)
We can see we are missing an x to the first power term in our dividend. We can write in 0x as a placeholder:
(x3 - 14x2 + 0x + 100) ÷ (x - 5)
Now we can perform our long division:
We can state our answer as: $$x^2 - 9x - 45 + \frac{-125}{x - 5}$$ The same process is used when our divisor is missing a term. Let's look at an example. Example 2: Find each quotient.
(x4 + 3x3 - 3x2 + 12x - 28) ÷ (x2 + 4)
Each polynomial is already in standard form. Our divisor is missing a term of x to the first power. We can write in 0x as a placeholder:
(x4 + 3x3 - 3x2 + 12x - 28) ÷ (x2 + 0x + 4)
Now we can perform our long division:
We can state our answer as: $$x^2 + 3x - 7$$ Skills Check:
Example #1
Find each quotient. $$\frac{x^{5}- 6x^{4}+ 6x - 36}{x - 6}$$
Please choose the best answer.
A
$$x^{4}+ 6$$
B
$$x^{4}+ 6 - \frac{5}{x - 6}$$
C
$$x^{4}+ 5 + \frac{4}{x - 6}$$
D
$$x^{4}+ 3 - \frac{1}{x - 6}$$
E
$$x^{4}+ 12$$
Example #2
Find each quotient. $$\frac{8x^{4}- 64x^{3}- 4x + 32}{x - 8}$$
Please choose the best answer.
A
$$8x^{3}- 4$$
B
$$8x^{3}+ 4$$
C
$$8x^{3}- 7 + \frac{4}{x - 8}$$
D
$$8x^{3}- 2 - \frac{3}{x - 8}$$
E
$$8x^{3}- 4 + \frac{4}{x - 8}$$
Example #3
Find each quotient. $$\frac{x^{3}+ 10x^{2}}{x + 10}$$
Please choose the best answer.
A
$$x^{2}+ \frac{1}{x + 10}$$
B
$$x^{2}+ 3x + \frac{1}{x + 10}$$
C
$$x^{2}$$
D
$$x^{2}+ 4$$
E
$$x^{2}+ \frac{3}{x + 10}$$
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