Operations with Polynomials

 

Learning Objective(s)

·         Add polynomials with more than one variable.

·         Subtract polynomials with more than one variable.

·         Multiply polynomials with more than one variable.

·         Divide polynomials with more than one variable.

 

 

Just as you can perform the four operations on polynomials with one variable, you can add, subtract, multiply, and divide polynomials with more than one variable. The process is exactly the same, but you have more variables to keep track of. When you are adding and subtracting polynomials with more than one variable, you have to pay particular care to combining like terms only. When you multiply and divide, you also need to pay particular attention to the multiple variables and terms. You can multiply and divide terms that aren’t like, but to add and subtract terms they must be like terms.

 

 

To add polynomials, you first need to identify the like terms in the polynomials and then combine them according to the correct integer operations. Since like terms must have the same exact variables raised to the same exact power, identifying them in polynomials with more than one variable takes a careful eye. Sometimes parentheses are used to distinguish between the addition of two polynomials and the addition of a collection of monomials. With addition, you can simply remove the parentheses and perform the addition.

 

 

 

 

Some people find that writing the polynomial addition in a vertical form makes it easy to combine like terms. The process of adding the polynomials is the same, but the arrangement of the terms is different. The example below shows this “vertical” method of adding polynomials:

 

 

 

 

When there isn't a matching like term for every term in each polynomial, there will be empty places in the vertical arrangement of the polynomials. This layout makes it easy to check that you are combining like terms only.

 

 

 

 

Add. (8a3b2 + 6a2b – 4b2 + 5) + (10a2b – 4a3b2 + 6a2 – 7)   A) 18a3b2 + 2a2b + 2b2 – 2   B) 4a3b2 + 16a2b + 6a2 – 4b2 – 2   C) 18a3b2 + 2a2b + 6a2 – 4b2 – 2   D) 4a6b4 + 16a4b2 + 6a2 – 4b2 – 2   Show/Hide Answer A) 18a3b2 + 2a2b + 2b2 – 2 Incorrect. Be sure to combine like terms only. The sequence of terms in the polynomials is not the same. Also, the terms −4b2 and 6a2 are not like terms and cannot be combined. The correct answer is 4a3b2 + 16a2b + 6a2 – 4b2 – 2.   B) 4a3b2 + 16a2b + 6a2 – 4b2 – 2 Correct. (8a3b2 + 6a2b – 4b2 + 5) + (10a2b – 4a3b2 + 6a2 – 7)   = 8a3b2 – 4a3b2 + 6a2b + 10a2b – 4b2 + 6a2 – 7+ 5   = 4a3b2 + 16a2b + 6a2 – 4b2 – 2.   C) 18a3b2 + 2a2b + 6a2 – 4b2 – 2 Incorrect. Be sure to combine like terms only. The sequence of terms in the polynomials is not the same. The correct answer is 4a3b2 + 16a2b + 6a2 – 4b2 – 2.   D) 4a6b4 + 16a4b2 + 6a2 – 4b2 – 2 Incorrect. When you add like terms, do not add the exponents, only the coefficients. The correct answer is 4a3b2 + 16a2b + 6a2 – 4b2 – 2.

 

 

 

You can apply the same process used to subtract polynomials with one variable to subtract polynomials with more than one variable. In order to remove the parentheses following a subtraction sign, you must multiply each term by −1.

 

 

 

 

An alternative to the approach shown above is the vertical method for arranging the subtraction problem. This method is shown below for a different problem. Both methods are effective for subtracting polynomials—the idea is to identify and organize like terms in order to compute with them accurately.

 

 

 

The examples that follow illustrate the left-to-right and vertical methods for the same polynomial subtraction problem. Think about which method you find easier.

 

 

 

 

 

 

 

Polynomials with more than one variable can also be multiplied by one another. You use the same techniques you used when you multiplied polynomials with only one variable. Consider the following example.

 

(4x²y³)(5x⁴y²)

 

This is an example of multiplication of two polynomials, specifically monomials, with two variables. To do this multiplication, you multiply the coefficients and use the rules of exponents to find the exponent for each variable in order to find the product. Let’s look.

 

(4x²y³)(5x⁴y²) = (4 • 5)(x²⁺⁴)(y³⁺²) = 20x⁶y⁵

 

To multiply a monomial by a binomial, you use the distributive property in the same way as multiplying polynomials with one variable.

 

 

 

 

To multiply two binomials containing more than one variable, you can still use the FOIL (First, Outer, Inner, Last) method that works for binomials with one variable. After all, FOIL is simply a shortcut for using the distributive property to multiply each term in one binomial by each term in the other binomial. This process works for multiplying any two binomials. Two examples follow.

 

 

 

 

The next example shows the product of a binomial and a trinomial, each with two variables. Since FOIL can only be used with the product of two binomials, you need to systematically multiply each term in the binomial by each term in the trinomial.

 

 

 

 

When multiplying multivariable polynomials like this, some people prefer to set up the multiplication in a vertical fashion—as you would do if you were multiplying 45 • 189. The example below shows (9b – ab)(5a²b + 7ab – b) set up in vertical fashion.

 

 

 

 

Notice that the products of the two examples are the same, although the order of the individual terms is different due to the different solution methods.

 

 

Find the product. −4pt2(5pt3 + 3pt2 – t)   A) −20p2t5 – 12p2t4 + 4pt3   B) −20t5 + 12p2t4 – 4pt3   C) −20pt6 – 12pt4 + 4pt2   D) −20p2t5 + 3pt2 – t   Show/Hide Answer A) −20p2t5 – 12p2t4 + 4pt3 Correct. Rewriting the subtraction as adding the opposite gives −4pt2(5pt3 + 3pt2 + (−t)). Distributing the monomial −4pt2 gives −4pt2 • 5pt3 + (−4pt2 • 3pt2) + (−4pt2 • −t), which is −20p2t5 – 12p2t4 + 4pt3.   B) −20t5 + 12p2t4 – 4pt3 Incorrect. The negative must be distributed to all terms along with the 4pt2. This changes the sign of the middle and last terms. The correct answer is −20p2t5 – 12p2t4 + 4pt3.   C) −20pt6 – 12pt4 + 4pt2 Incorrect. By the laws of exponents, you add (not multiply) exponents when multiplying: −4pt2 • 5pt3 + (−4pt2 • 3pt2) + (−4pt2 • −t) is −20p2t5 – 12p2t4 + 4pt3.The correct answer is −20p2t5 – 12p2t4 + 4pt3.   D) −20p2t5 + 3pt2 – t Incorrect. You must distribute the monomial to all three terms in the polynomial, not just the first one: −4pt2 • 5pt3 + (−4pt2 • 3pt2) + (−4pt2 • −t). The correct answer is −20p2t5 – 12p2t4 + 4pt3.

 

 

 

The fourth arithmetic operation is division. Polynomials with more than one variable can also be divided. When dividing monomials with more than one variable, you divide the coefficients and then divide variables. When there are exponents with the same base, the law of exponents says you divide by subtracting the exponents. Consider this example.

 

 

Now let’s look at an example of dividing a trinomial with more than one variable by a monomial with more than one variable. This follows the same procedure as when you have one variable, but you need to pay attention to distinguishing between the variables.

 

 

 

 

 

Divide.   A) 5st2 – 2s2t + 1   B) 5st2 – 10s3t3 + 5st2   C) 20st2 – 5s2t   D)   Show/Hide Answer A) 5st2 – 2s2t + 1 Correct. Divide each term in the polynomial by the monomial: .   B) 5st2 – 10s3t3 + 5st2 Incorrect. You only divided the first term. Divide each term in the polynomial by the monomial: . The correct answer is 5st2 – 2s2t + 1.   C) 20st2 – 5s2t Incorrect. Divide, don’t subtract, the coefficients The correct answer is 5st2 – 2s2t + 1.   D) Incorrect. Perform the division to simplify. It’s okay that they are not like terms, you can still divide. You have to have like terms when you add or subtract terms, not when you multiply and divide terms. Divide each term in the polynomial by the monomial: . The correct answer is 5st2 – 2s2t + 1.

 

 

 

Performing addition, subtraction, multiplication, and division of polynomials with more than one variable follows the same steps as operating on polynomials in one variable. The key things to pay attention to are combining only like terms and applying the laws of exponents, integer operations, and the order of operations accurately.