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Study Guides > Intermediate Algebra

Divide Polynomials Part I

Learning Outcome

  • Multiply and divide polynomials

Divide a Polynomial by a Binomial

Dividing a polynomial by a monomial can be handled by dividing each term in the polynomial separately. This cannot be done when the divisor has more than one term. However, the process of long division can be very helpful with polynomials. Recall how you can use long division to divide two whole numbers, say 900900 divided by 3737. The dividend in 900 and the divisor is 37. First, you would think about how many 37s37s are in 9090, as 99 is too small. (Note: you could also think, how many 40s40s are there in 9090.) There are two 37s37s in 9090, so write 22 above the last digit of 9090. Two 37s37s is 7474; write that product below the 9090. Screen Shot 2016-03-28 at 3.35.17 PM Subtract: 907490–74 is 1616. (If the result is larger than the divisor, 3737, then you need to use a larger number for the quotient.) Bring down the next digit (0)(0) and consider how many 37s37s are in 160160. Screen Shot 2016-03-28 at 3.36.06 PM There are four 37s37s in 160160, so write the 44 next to the two in the quotient. Four 37s37s is 148148; write that product below the 160160. Subtract: 160148160–148 is 1212. This is less than 3737 so the 44 is correct. Since there are no more digits in the dividend to bring down, you are done. The final answer is 2424 R1212, or 24123724\frac{12}{37}. You can check this by multiplying the quotient (without the remainder) by the divisor, and then adding in the remainder. The result should be the dividend:

2437+12=888+12=90024\cdot37+12=888+12=900

To divide polynomials, use the same process. This example shows how to do this when dividing by a binomial.

Example

Divide: (x24x12)(x+2)\frac{\left(x^{2}–4x–12\right)}{\left(x+2\right)}

Answer: How many x’s are there in x2x^{2}? That is, what is x2x \frac{{{x}^{2}}}{x}? x2x=x \frac{{{x}^{2}}}{x}=x. Put x in the quotient above the 4x-4x term. These are like terms, which helps to organize the problem. Write the product of the divisor and the part of the quotient you just found under the dividend. Since x(x+2)=x2+2xx\left(x+2\right)=x^{2}+2x, write this underneath, and get ready to subtract. Rewrite (x2+2x)–\left(x^{2} + 2x\right) as its opposite x22x–x^{2}–2x so that you can add the opposite. Adding the opposite is the same as subtracting, and it is easier to do. Add x2-x^{2} to x2x^{2} and 2x-2x to 4x-4x. Bring down 12-12. Screen Shot 2016-03-28 at 4.19.50 PM Repeat the process. How many times does x go into 6x-6x? In other words, what is 6xx \frac{-6x}{x}? Since 6xx=6 \frac{-6x}{x}=-6, write 6-6 in the quotient. Multiply 6-6 and x+2x+2 and prepare to subtract the product. Screen Shot 2016-03-28 at 4.24.52 PM Rewrite (6x12)–\left(-6x–12\right) as 6x+126x+12, so that you can add the opposite. Add. In this case, there is no remainder, so you are done. The answer is x6x–6. Check this by multiplying:

(x6)(x+2)=x2+2x6x12=x24x12\left(x-6\right)\left(x+2\right)=x^{2}+2x-6x-12=x^{2}-4x-12

In the following video, we show another example of dividing a degree two trinomial by a degree one binomial. https://youtu.be/KUPFg__Djzw Let us try another example. In this example, a term is “missing” from the dividend.

Example

Divide: (x36x10)(x3)\frac{\left(x^{3}–6x–10\right)}{\left(x–3\right)}

Answer: In setting up this problem, notice that there is an x3x^{3} term but no x2x^{2} term. Add 0x20x^{2} as a “place holder” for this term. Since 0 times anything is 0, you are not changing the value of the dividend. Focus on the first terms again: how many x’s are there in x3x^{3}? Since x3x=x2 \frac{{{x}^{3}}}{x}=x^{2}, put x2x^{2} in the quotient. Multiply: x2(x3)=x33x2x^{2}\left(x–3\right)=x^{3}–3x^{2}; write this underneath the dividend, and prepare to subtract. Rewrite the subtraction using the opposite of the expression x33x2x^{3}-3x^{2}. Then add. Bring down the rest of the expression in the dividend. It is helpful to bring down all of the remaining terms. Now, repeat the process with the remaining expression, 3x26x103x^{2}-6x–10, as the dividend. Remember to watch the signs! How many x’s are there in 33x? Since there are 33, multiply 3(x3)=3x93\left(x–3\right)=3x–9, write this underneath the dividend, and prepare to subtract. Continue until the degree of the remainder is less than the degree of the divisor. In this case the degree of the remainder, 1-1, is 00, which is less than the degree of x3x-3, which is 11. Also notice that you have brought down all the terms in the dividend and that the quotient extends to the right edge of the dividend. These are other ways to check whether you have completed the problem. You can write the remainder using the symbol R or as a fraction added to the rest of the quotient with the remainder in the numerator and the divisor in the denominator. In this case, since the remainder is negative, you can also subtract the opposite. Possible forms of the solution are: x2+3x+3+R1,x2+3x+3+1x3, or x2+3x+31x3\begin{array}{l}x^{2}+3x+3+R-1,\\x^{2}+3x+3+\frac{-1}{x-3}, \text{ or }\\x^{2}+3x+3-\frac{1}{x-3}\end{array} Check the result: (x3)(x2+3x+3)=x(x2+3x+3)3(x2+3x+3)=x3+3x2+3x3x29x9=x36x9\begin{array}{l}\left(x–3\right)\left(x^{2}+3x+3\right) & =x\left(x^{2}+3x+3\right)–3\left(x^{2}+3x+3\right)\\ & =x^{3}+3x^{2}+3x–3x^{2}–9x–9\\ & =x^{3}–6x–9\end{array} x36x9+(1)=x36x10x^{3}–6x–9+\left(-1\right)=x^{3}–6x–10

In the video that follows, we show another example of dividing a degree three trinomial by a binomial. Note the "missing" term and how we work with it. https://youtu.be/Rxds7Q_UTeo The process above works for dividing any polynomials no matter how many terms are in the divisor or the dividend. The main things to remember are:
  • When subtracting, be sure to subtract the whole expression, not just the first term. This is very easy to forget, so be careful!
  • Stop when the degree of the remainder is less than the degree of the dividend or when you have brought down all the terms in the dividend and your quotient extends to the right edge of the dividend.
In the next video, we present one more example of polynomial long division. https://youtu.be/P6OTbUf8f60

Summary

Dividing polynomials by polynomials of more than one term can be done using a process very much like long division of whole numbers. You must be careful to subtract entire expressions, not just the first term. Stop when the degree of the remainder is less than the degree of the divisor. The remainder can be written using R notation or as a fraction added to the quotient with the remainder in the numerator and the divisor in the denominator.

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