Lesson Objectives
- Demonstrate an understanding of how to divide a polynomial by a monomial
- Demonstrate an understanding of how to write a polynomial in standard form
- Demonstrate an understanding of the acronym DMSBR for long division
- Learn how to divide polynomials using long division
- Learn how to check the answer from a polynomial long division
How to Divide Polynomials Using Long Division
In our last lesson, we learned how to divide a polynomial by a monomial. In this lesson, we will go one step further and learn how to divide polynomials using long division. Polynomial long division is a step-by-step process that is very similar to long division with whole numbers. Recall the DMSBR acronym shows the steps of the long division process:
D » Divide
M » Multiply
S » Subtract
B » Bring Down
R » Repeat or Remainder
Let's work through some examples.
Example 1: Find each quotient.
(x3 + 6x2 + 7x + 2) ÷ (x + 1)
Before we begin our long division, we need to place both dividend and divisor in standard form. We also want to make sure that there are no missing powers. This means if our highest power on x is a 4, we will have x to the 3rd power, 2nd power, and 1st power as well. In this case, each polynomial is already in standard form and we have no missing powers.
Let's begin by placing our dividend (x3 + 6x2 + 7x + 2) under the long division symbol and our divisor (x + 1) to the left of the long division symbol:
DMSBR:
Step 1) Divide » This is slightly different than working with whole numbers. Here we only divide leading term by leading term:
Our leading term of the dividend is (x3) and our leading term of our divisor is (x). We perform the following division: $$\frac{x^3}{x}=x^{3 - 1}=x^2$$ When we write our answer from a division, we want to maintain place value. Since our exponent on x is a 2, we want to place our answer over the 6x2: 
DMSBR:
Step 2) Multiply » We will multiply our answer (x2) by our divisor (x + 1). We place the result under the dividend:
We will perform the following multiplication: $$x^2(x + 1)=x^3 + x^2$$ Now we can place our answer (x3 + x2) under the dividend: 
DMSBR:
Step 3) Subtract » We will subtract our answer from the multiplication away from our dividend. We enclose our subtrahend in parentheses to remind us we are subtracting away each term:
To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(x^3 + x^2)=-x^3 - x^2$$ 
Now we can combine like terms: $$x^3 - x^3=0$$ $$6x^2 - 1x^2=5x^2$$ 
DMSBR:
Step 4) Bring Down » We will bring down the next term from the dividend (7x):
DMSBR:
Step 5) Repeat or Remainder » In this case, we brought down a term (7x) from the dividend. This means we will repeat our steps and start with division again.
DMSBR:
Step 1) Divide » Divide leading term by leading term:
Our leading term of the dividend is (5x2) and our leading term of our divisor is (x). We perform the following division: $$\frac{5x^2}{x}=5x^{2 - 1}=5x$$ Since our exponent on x is a 1, we want to place our answer over the 7x: 
DMSBR:
Step 2) Multiply » We will multiply our answer (5x) by our divisor (x + 1). We place the result under the dividend:
We will perform the following multiplication: $$5x(x + 1)=5x^2 + 5x$$ Now we can place our answer (5x2 + 5x) under the dividend: 
DMSBR:
Step 3) Subtract » We will subtract our answer from the multiplication away from our dividend. We enclose our subtrahend in parentheses to remind us we are subtracting away each term:
To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(5x^2 + 5x)=-5x^2 - 5x$$ 
Now we can combine like terms: $$5x^2 - 5x^2=0$$ $$7x - 5x=2x$$ 
DMSBR:
Step 4) Bring Down » We will bring down the next term from the dividend (2):
DMSBR:
Step 5) Repeat or Remainder » In this case, we brought down a term (2) from the dividend. This means we will repeat our steps and start with division again.
DMSBR:
Step 1) Divide » Divide leading term by leading term:
Our leading term of the dividend is (2x) and our leading term of our divisor is (x). We perform the following division: $$\frac{2x}{x}=2$$ Since we have a constant, we want to place our answer over the 2: 
DMSBR:
Step 2) Multiply » We will multiply our answer (2) by our divisor (x + 1). We place the result under the dividend:
We will perform the following multiplication: $$2(x + 1)=2x + 2$$ Now we can place our answer (2x + 2) under the dividend: 
DMSBR:
Step 3) Subtract » We will subtract our answer from the multiplication away from our dividend. We enclose our subtrahend in parentheses to remind us we are subtracting away each term:
To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(2x + 2)=-2x - 2$$ 
Now we can combine like terms: $$2x - 2x=0$$ $$2 - 2=0$$ 
DMSBR:
Step 4) Bring Down » There are no more terms to bring down.
DMSBR:
Step 5) Repeat or Remainder » Since we didn't bring anything down in our last step and the result of our last subtraction was zero, we have no remainder. We can report our answer as:
x2 + 5x + 2
We can check this result by multiplying by the divisor (x + 1). This will give us our dividend (x3 + 6x2 + 7x + 2) as a result: $$(x+1)(x^2+5x+2)$$ $$x(x^2+5x+2) + 1(x^2 + 5x + 2)$$ $$x^3 + 5x^2 + 2x + x^2 + 5x + 2$$ $$x^3 + 6x^2 + 7x + 2$$ We can see that we got our dividend back, so our solution is correct. Let's take a look at an example where a remainder is involved.
Example 2: Find each quotient.
(7x2 - 20x + 3) ÷ (x - 2)
Before we begin our long division, we need to place both dividend and divisor in standard form. We also want to make sure that there are no missing powers. In this case, each polynomial is already in standard form and we have no missing powers.
Let's begin by placing our dividend (7x2 - 20x + 3) under the long division symbol and our divisor (x - 2) to the left of the long division symbol:
DMSBR:
Step 1) Divide » Divide leading term by leading term:
Our leading term of the dividend is (7x2) and our leading term of our divisor is (x). We perform the following division: $$\frac{7x^2}{x}=7x^{2 - 1}=7x$$ Since our exponent on x is a 1, we want to place our answer over the 20x: 
DMSBR:
Step 2) Multiply » We will multiply our answer (7x) by our divisor (x - 2). We place the result under the dividend:
We will perform the following multiplication: $$7x(x - 2)=7x^2 - 14x$$ Now we can place our answer (7x2 - 14x) under the dividend: 
DMSBR:
Step 3) Subtract » We will subtract our answer from the multiplication away from our dividend. We enclose our subtrahend in parentheses to remind us we are subtracting away each term:
To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(7x^2 - 14x)=-7x^2 + 14x$$ 
Now we can combine like terms: $$7x^2 - 7x^2=0$$ $$-20x + 14x=-6x$$ 
DMSBR:
Step 4) Bring Down » We will bring down the next term from the dividend (3):
DMSBR:
Step 5) Repeat or Remainder » In this case, we brought down a term (3) from the dividend. This means we will repeat our steps and start with division again.
DMSBR:
Step 1) Divide » Divide leading term by leading term:
Our leading term of the dividend is (-6x) and our leading term of our divisor is (x). We perform the following division: $$\frac{-6x}{x}=-6$$ Since we have a constant, we want to place our answer over the 3: 
DMSBR:
Step 2) Multiply » We will multiply our answer (-6) by our divisor (x - 2). We place the result under the dividend:
We will perform the following multiplication: $$-6(x - 2)=-6x + 12$$ Now we can place our answer (-6x + 12) under the dividend: 
DMSBR:
Step 3) Subtract » We will subtract our answer from the multiplication away from our dividend. We enclose our subtrahend in parentheses to remind us we are subtracting away each term:
To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(-6x + 12)=6x - 12$$ 
Now we can combine like terms: $$-6x + 6x=0$$ $$3 - 12=-9$$ 
DMSBR:
Step 4) Bring Down » There are no more terms to bring down.
DMSBR:
Step 5) Repeat or Remainder » Since we didn't bring anything down in our last step and the result of our last subtraction was not zero, we have a remainder. When we have a remainder, we add the remainder over the divisor to the quotient. We can report our answer as: $$7x - 6 + \frac{-9}{x - 2}$$ We can check this result by multiplying by the divisor (x - 2). This will give us our dividend (7x2 - 20x + 3) as a result: $$(x - 2) \left(7x - 6 + \frac{-9}{x - 2}\right)$$ Let's distribute the (x - 2) to each term inside of the parentheses: $$7x(x-2)=7x^2 - 14x$$ $$-6(x-2)=-6x + 12$$ $$(x-2)\left(\frac{-9}{x-2}\right)=-9$$ Now let's combine like terms: $$7x^2 - 14x - 6x + 12 - 9$$ $$7x^2 - 20x + 3$$ We can see that we got our dividend back, so our solution is correct.
D » Divide
M » Multiply
S » Subtract
B » Bring Down
R » Repeat or Remainder
Let's work through some examples.
Example 1: Find each quotient.
(x3 + 6x2 + 7x + 2) ÷ (x + 1)
Before we begin our long division, we need to place both dividend and divisor in standard form. We also want to make sure that there are no missing powers. This means if our highest power on x is a 4, we will have x to the 3rd power, 2nd power, and 1st power as well. In this case, each polynomial is already in standard form and we have no missing powers.
Let's begin by placing our dividend (x3 + 6x2 + 7x + 2) under the long division symbol and our divisor (x + 1) to the left of the long division symbol:
DMSBR: Step 1) Divide » This is slightly different than working with whole numbers. Here we only divide leading term by leading term:
Our leading term of the dividend is (x3) and our leading term of our divisor is (x). We perform the following division: $$\frac{x^3}{x}=x^{3 - 1}=x^2$$ When we write our answer from a division, we want to maintain place value. Since our exponent on x is a 2, we want to place our answer over the 6x2: DMSBR: Step 2) Multiply » We will multiply our answer (x2) by our divisor (x + 1). We place the result under the dividend:
We will perform the following multiplication: $$x^2(x + 1)=x^3 + x^2$$ Now we can place our answer (x3 + x2) under the dividend: DMSBR: Step 3) Subtract » We will subtract our answer from the multiplication away from our dividend. We enclose our subtrahend in parentheses to remind us we are subtracting away each term:
To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(x^3 + x^2)=-x^3 - x^2$$ Now we can combine like terms: $$x^3 - x^3=0$$ $$6x^2 - 1x^2=5x^2$$ DMSBR: Step 4) Bring Down » We will bring down the next term from the dividend (7x):
DMSBR: Step 5) Repeat or Remainder » In this case, we brought down a term (7x) from the dividend. This means we will repeat our steps and start with division again.
DMSBR:
Step 1) Divide » Divide leading term by leading term:
Our leading term of the dividend is (5x2) and our leading term of our divisor is (x). We perform the following division: $$\frac{5x^2}{x}=5x^{2 - 1}=5x$$ Since our exponent on x is a 1, we want to place our answer over the 7x: DMSBR: Step 2) Multiply » We will multiply our answer (5x) by our divisor (x + 1). We place the result under the dividend:
We will perform the following multiplication: $$5x(x + 1)=5x^2 + 5x$$ Now we can place our answer (5x2 + 5x) under the dividend: DMSBR: Step 3) Subtract » We will subtract our answer from the multiplication away from our dividend. We enclose our subtrahend in parentheses to remind us we are subtracting away each term:
To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(5x^2 + 5x)=-5x^2 - 5x$$ Now we can combine like terms: $$5x^2 - 5x^2=0$$ $$7x - 5x=2x$$ DMSBR: Step 4) Bring Down » We will bring down the next term from the dividend (2):
DMSBR: Step 5) Repeat or Remainder » In this case, we brought down a term (2) from the dividend. This means we will repeat our steps and start with division again.
DMSBR:
Step 1) Divide » Divide leading term by leading term:
Our leading term of the dividend is (2x) and our leading term of our divisor is (x). We perform the following division: $$\frac{2x}{x}=2$$ Since we have a constant, we want to place our answer over the 2: DMSBR: Step 2) Multiply » We will multiply our answer (2) by our divisor (x + 1). We place the result under the dividend:
We will perform the following multiplication: $$2(x + 1)=2x + 2$$ Now we can place our answer (2x + 2) under the dividend: DMSBR: Step 3) Subtract » We will subtract our answer from the multiplication away from our dividend. We enclose our subtrahend in parentheses to remind us we are subtracting away each term:
To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(2x + 2)=-2x - 2$$ Now we can combine like terms: $$2x - 2x=0$$ $$2 - 2=0$$ DMSBR: Step 4) Bring Down » There are no more terms to bring down.
DMSBR:
Step 5) Repeat or Remainder » Since we didn't bring anything down in our last step and the result of our last subtraction was zero, we have no remainder. We can report our answer as:
x2 + 5x + 2
We can check this result by multiplying by the divisor (x + 1). This will give us our dividend (x3 + 6x2 + 7x + 2) as a result: $$(x+1)(x^2+5x+2)$$ $$x(x^2+5x+2) + 1(x^2 + 5x + 2)$$ $$x^3 + 5x^2 + 2x + x^2 + 5x + 2$$ $$x^3 + 6x^2 + 7x + 2$$ We can see that we got our dividend back, so our solution is correct. Let's take a look at an example where a remainder is involved.
Example 2: Find each quotient.
(7x2 - 20x + 3) ÷ (x - 2)
Before we begin our long division, we need to place both dividend and divisor in standard form. We also want to make sure that there are no missing powers. In this case, each polynomial is already in standard form and we have no missing powers.
Let's begin by placing our dividend (7x2 - 20x + 3) under the long division symbol and our divisor (x - 2) to the left of the long division symbol:
DMSBR: Step 1) Divide » Divide leading term by leading term:
Our leading term of the dividend is (7x2) and our leading term of our divisor is (x). We perform the following division: $$\frac{7x^2}{x}=7x^{2 - 1}=7x$$ Since our exponent on x is a 1, we want to place our answer over the 20x: DMSBR: Step 2) Multiply » We will multiply our answer (7x) by our divisor (x - 2). We place the result under the dividend:
We will perform the following multiplication: $$7x(x - 2)=7x^2 - 14x$$ Now we can place our answer (7x2 - 14x) under the dividend: DMSBR: Step 3) Subtract » We will subtract our answer from the multiplication away from our dividend. We enclose our subtrahend in parentheses to remind us we are subtracting away each term:
To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(7x^2 - 14x)=-7x^2 + 14x$$ Now we can combine like terms: $$7x^2 - 7x^2=0$$ $$-20x + 14x=-6x$$ DMSBR: Step 4) Bring Down » We will bring down the next term from the dividend (3):
DMSBR: Step 5) Repeat or Remainder » In this case, we brought down a term (3) from the dividend. This means we will repeat our steps and start with division again.
DMSBR:
Step 1) Divide » Divide leading term by leading term:
Our leading term of the dividend is (-6x) and our leading term of our divisor is (x). We perform the following division: $$\frac{-6x}{x}=-6$$ Since we have a constant, we want to place our answer over the 3: DMSBR: Step 2) Multiply » We will multiply our answer (-6) by our divisor (x - 2). We place the result under the dividend:
We will perform the following multiplication: $$-6(x - 2)=-6x + 12$$ Now we can place our answer (-6x + 12) under the dividend: DMSBR: Step 3) Subtract » We will subtract our answer from the multiplication away from our dividend. We enclose our subtrahend in parentheses to remind us we are subtracting away each term:
To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(-6x + 12)=6x - 12$$ Now we can combine like terms: $$-6x + 6x=0$$ $$3 - 12=-9$$ DMSBR: Step 4) Bring Down » There are no more terms to bring down.
DMSBR:
Step 5) Repeat or Remainder » Since we didn't bring anything down in our last step and the result of our last subtraction was not zero, we have a remainder. When we have a remainder, we add the remainder over the divisor to the quotient. We can report our answer as: $$7x - 6 + \frac{-9}{x - 2}$$ We can check this result by multiplying by the divisor (x - 2). This will give us our dividend (7x2 - 20x + 3) as a result: $$(x - 2) \left(7x - 6 + \frac{-9}{x - 2}\right)$$ Let's distribute the (x - 2) to each term inside of the parentheses: $$7x(x-2)=7x^2 - 14x$$ $$-6(x-2)=-6x + 12$$ $$(x-2)\left(\frac{-9}{x-2}\right)=-9$$ Now let's combine like terms: $$7x^2 - 14x - 6x + 12 - 9$$ $$7x^2 - 20x + 3$$ We can see that we got our dividend back, so our solution is correct.
Skills Check:
Example #1
Find each quotient. $$\frac{15x^{2}+ 47x + 28}{3x + 4}$$
Please choose the best answer.
A
$$5x + 12 - \frac{4}{3x + 4}$$
B
$$5x + 9 - \frac{8}{3x + 4}$$
C
$$5x + 10 - \frac{7}{3x + 4}$$
D
$$5x + 11 - \frac{10}{3x + 4}$$
E
$$5x + 1$$
Example #2
Find each quotient. $$\frac{28x^{2}+ 7x - 44}{7x - 7}$$
Please choose the best answer.
A
$$6x + 2 - \frac{2}{x - 1}$$
B
$$6x + 8 - \frac{11}{7x - 7}$$
C
$$4x + 5 + \frac{10}{7x - 7}$$
D
$$4x - 5 - \frac{9}{x - 1}$$
E
$$4x + 5 - \frac{9}{7x - 7}$$
Example #3
Find each quotient. $$\frac{24x^{3}- 2x^{2}- 16x - 32}{6x - 8}$$
Please choose the best answer.
A
$$4x^{2}+ 5x + 3 + \frac{1}{6x - 8}$$
B
$$4x^{2}+ 9x + 1 + \frac{7}{6x - 8}$$
C
$$4x^{2}- 11x + 7$$
D
$$4x^{2}+ 7x - 6$$
E
$$4x^{2}+ 5x + 4$$
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