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: polynomial long division set up DMSBR:
Step 1) Divide » This is slightly different than working with whole numbers. Here we only divide leading term by leading term: polynomial long division step 1 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: polynomial long division write the answer according to place value DMSBR:
Step 2) Multiply » We will multiply our answer (x2) by our divisor (x + 1). We place the result under the dividend: polynomial long division multiplication step 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: polynomial long division multiplication step 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: polynomial long division subtraction step To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(x^3 + x^2) = -x^3 - x^2$$ polynomial long division subtraction step Now we can combine like terms: $$x^3 - x^3 = 0$$ $$6x^2 - 1x^2 = 5x^2$$ polynomial long division subtraction step DMSBR:
Step 4) Bring Down » We will bring down the next term from the dividend (7x): polynomial long division bring down step 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: polynomial long division step 1 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: polynomial long division write the answer according to place value DMSBR:
Step 2) Multiply » We will multiply our answer (5x) by our divisor (x + 1). We place the result under the dividend: polynomial long division multiplication step We will perform the following multiplication: $$5x(x + 1) = 5x^2 + 5x$$ Now we can place our answer (5x2 + 5x) under the dividend: polynomial long division multiplication step 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: polynomial long division subtraction step To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(5x^2 + 5x) = -5x^2 - 5x$$ polynomial long division subtraction step Now we can combine like terms: $$5x^2 - 5x^2 = 0$$ $$7x - 5x = 2x$$ polynomial long division subtraction step DMSBR:
Step 4) Bring Down » We will bring down the next term from the dividend (2): polynomial long division bring down step 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: polynomial long division step 1 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: polynomial long division write the answer according to place value DMSBR:
Step 2) Multiply » We will multiply our answer (2) by our divisor (x + 1). We place the result under the dividend: polynomial long division multiplication step We will perform the following multiplication: $$2(x + 1) = 2x + 2$$ Now we can place our answer (2x + 2) under the dividend: polynomial long division multiplication step 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: polynomial long division subtraction step To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(2x + 2) = -2x - 2$$ polynomial long division subtraction step Now we can combine like terms: $$2x - 2x = 0$$ $$2 - 2 = 0$$ polynomial long division subtraction step 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: polynomial long division set up DMSBR:
Step 1) Divide » Divide leading term by leading term: polynomial long division step 1 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: polynomial long division write the answer according to place value DMSBR:
Step 2) Multiply » We will multiply our answer (7x) by our divisor (x - 2). We place the result under the dividend: polynomial long division multiplication step We will perform the following multiplication: $$7x(x - 2) = 7x^2 - 14x$$ Now we can place our answer (7x2 - 14x) under the dividend: polynomial long division multiplication step 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: polynomial long division subtraction step To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(7x^2 - 14x) = -7x^2 + 14x$$ polynomial long division subtraction step Now we can combine like terms: $$7x^2 - 7x^2 = 0$$ $$-20x + 14x = -6x$$ polynomial long division subtraction step DMSBR:
Step 4) Bring Down » We will bring down the next term from the dividend (3): polynomial long division bring down step 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: polynomial long division step 1 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: polynomial long division write the answer according to place value DMSBR:
Step 2) Multiply » We will multiply our answer (-6) by our divisor (x - 2). We place the result under the dividend: polynomial long division multiplication step We will perform the following multiplication: $$-6(x - 2) = -6x + 12$$ Now we can place our answer (-6x + 12) under the dividend: polynomial long division multiplication step 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: polynomial long division subtraction step To perform the subtraction, we first distribute the negative to each term inside of parentheses: $$-(-6x + 12) = 6x - 12$$ polynomial long division subtraction step Now we can combine like terms: $$-6x + 6x = 0$$ $$3 - 12 = -9$$ polynomial long division subtraction step 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 - 20x2 + 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.