Idiot's Guides: Algebra I (2015)
Part IV. Polynomials
My detective stories start out with everything seeming clear, but before the matter gets resolved, you know it’s going to get complicated. What do the heroes do when things get complex? Usually, they step back and try to apply essential truths to the evidence they’ve gathered, and often they actually have to back up. They need to work backward to understand what they’re dealing with. Once they’ve navigated those complications, however, they always find the critical piece of evidence that unravels the entire mystery.
With Part 4, we’ve reached that point in our algebra story. Polynomials look complicated at first glance, but we’ll master them by applying crucial rules we learned in arithmetic. When we’ve learned to operate on those polynomials, we’ll do our backing up. We’ll factor polynomials, taking them back to the products that produced them. That will be the key that makes everything that follows fall into place. Isn’t that what always happens?
Chapter 11. Adding and Subtracting Polynomials
In This Chapter
· Identifying polynomials by size and degree
· Adding polynomials
· Subtracting polynomials by adding the opposite
· Simplifying expressions with addition and subtraction
In previous chapters, we’ve talked a lot about terms, expressions, and equations. That’s natural, because the language of algebra is built on those elements. In this chapter, and several that follow, we’re going to focus on certain types of terms, expressions, and equations. We’re going to bring our focus to a category of algebraic operations, working with expressions called polynomials. Much of what you’ve learned will carry over, but polynomials will present some new concerns. As is so often the case, we’ll begin with some new vocabulary.
Form and Degree
A monomial is a product of a real number, called the coefficient, and a variable raised to a non-negative integer power. Take that apart and you’ll see that monomial is the name given to the product of a coefficient and a power of a variable. Examples of monomials would include and 4t5.
A monomial is a constant, a variable, or a product of constants and variables.
If you’re thinking that sounds a lot like what we’ve been calling a term, you’re right. A monomial is a term, with a few additional properties. The exponents cannot be negative numbers and cannot be fractions. An exponent of 1 is fine, so – 3x is a monomial, and an exponent of 0 is acceptable, too, so a constant like – 7 = – 7 x0 also counts as a monomial. But if there’s a variable in a denominator, like , or under a radical, like , it’s not a monomial.
Most of the monomials you’ll encounter involve only one variable, although it is possible to have a monomial with more than one variable. The degree of a monomial is the power to which the variable is raised. is degree two, or second degree. 4t5 is fifth degree. −3x is first degree, degree 1, and constants are degree zero.
The degree of a monomial with one variable is the power of the variable. A constant has degree zero.
When you add or subtract monomials, you form expressions that take the general name polynomials. This is the general name because it applies to any expression that is made up of monomials, including just one monomial. A monomial is a polynomial, and any sum of monomials is a polynomial. 6x3 and – 9x8 are both monomials, and 6x3 + −9x8 or 6x3 – 9x8 is a polynomial. The expression is also a polynomial. The term polynomial applies no matter how many monomials are involved.
The polynomials you will work with most often generally have two or three monomials, and those common ones get their own names. A polynomial made of two monomials is a binomial, and one with three monomials is a trinomial. Anything bigger than that gets the general name polynomial.
A polynomial is a sum of monomials.
To determine the degree of a polynomial, look at the degree of each of the monomials that form the polynomial. The highest of those is the degree of the polynomial. is a sixth-degree polynomial and 6x3 + −9x8 is an eighth-degree binomial. You will most often see first-degree binomials like 7x − 3 and second-degree trinomials like x2 + 4x −9.
It’s easier to determine the degree of a polynomial if it’s in standard form. Standard form means that the monomials that make up the polynomial have been arranged in order, from highest degree to lowest. and x2 + 4x −9 are both polynomials in standard form. When a polynomial is in standard form, the degree of the polynomial is the degree of the first term.
Tell whether the expression is a polynomial.
2. 3y7 − 4y3 + 8
Put each polynomial in standard form.
4. 9z − 6z2 + 2
5. 4x2 − 3x + x3 − 2
6. 8t − 2 + t2
Identify each expression as a monomial, binomial, trinomial, or polynomial, and give its degree.
7. 9y − 3
8. x2 + 5x − 9
10. 12a9 + 8a12 − 4a7 + 6a4 + 2
Addition of Polynomials
You already know most of the rules you need to do arithmetic with polynomials. Because polynomials are a class of expressions, the rules you learned for operations with variables will apply.
To add two polynomials, you simply need to combine like terms. If you want to add the binomial −3x − 6 to the trinomial 4x2 − 7x + 12, your job will be to add the like terms and only the like terms. Both of these polynomials are in standard form, so it will be easy to write them one under another, with like terms aligned. This vertical arrangement makes addition easier.
From here, it’s basically the same process used in standard arithmetic. When you add two numbers, you first add the ones digits, then the tens digits, and so on, right to left. When you add two polynomials, you first add the constants, then the first degree terms, and so on, right to left. There’s no need to carry. Just add straight down.
The vertical arrangement is convenient when the polynomials are in standard form and particularly useful when adding more than two polynomials. When you add polynomials vertically, you just leave an empty space if a particular power is absent from a polynomial. If you add 3x6 + 9x4 − 7x3 + 2x2 − 1 and 4x5 − 3x4 + 11x2 –x + 8, you place them one under the other with terms that contain the same power of the variable aligned. Space the terms out whenever a power is missing, so that the terms in each column are like terms.
If you prefer, you can insert zeros to hold places for the missing powers.
Once the polynomials are arranged with like terms in columns, you add down the column. If there’s only one term in a column, you can carry it down.
If there are more than two polynomials to be added, you still add down each column, you just need a taller stack. To add 6x2 − 3x + 2, 9x2 − 7x + 1, and 4x2 + 9x − 3, stack them with like terms aligned.
The columns are taller, but the process is the same. Add two terms, and then add the third term to that total.
You can drop the +0 and just give the sum as 19x2 − x.
If the polynomials are not in standard form, or they’re not too complicated, you might prefer not to use a vertical arrangement, and just keep all your work on a line. If you do make that choice, organization will be important. Either take a moment to rearrange the terms so that like terms are grouped together, or cross off terms as you add them.
If you’re adding 4t3 − 7t + 9t2 + 2 to 6 − 2t + t3 − 8t4 + t2, you might want to organize the problem this way.
(4t3 − 7t + 9t2 + 2) + (6 − 2t + t3 − 8t4 + t2)
= −8t4 + (4t3 + t3) + (9t2 + t2) + (−7t − 2t) + (2 + 6)
= − 8t4 + 5t3 + 10t2 −9t + 8
The biggest danger in rearranging the terms of the polynomials is separating a number from its sign. Keep every sign firmly attached to a coefficient. If you move a term, make sure its sign moves with it.
If you don’t want to rewrite the problem to rearrange the terms, you can cross out terms to keep track of what you’ve done.
(4t3 − 7t + 9t2 + 2) + (6 − 2t + t3 − 8t4 + t2)
There’s only one fourth power term, so write that down and cross it out.
Add the third power terms and cross them out when you’re done.
Next the squares, then the first power terms.
Combine the constants and you’re done.
When every term is crossed out, you know your job is done.
Perform each addition. Use the method you think is most efficient.
11. (5 x3) + (7 x2) + (2x3)
12. (4x − 7) + (9 − 2x)
13. (3b + b2) + (5b − 6b2)
14. (3x2 − 2x − 5) + (−5x2 − 4x − 6)
15. (x2 − 3x + 4) + (−7x2 − 5x)
16. (x2 − 8) + (5x2 + 6x − 1)
17. (2y3 + 3y2 − y) + (5y3 − 6y2 + 3y) + (2y3 − y2 − 7y)
18. (2y2 + 6y − 5) + (7y − 3y2 + 6) + (8 − 9y + 2y2)
19. (3t2 − 2t + 6) + (7t − t2 − 1) + (4 − 6t2 + t)
20. (2x + 3x2) + (−5x3 − 4x) + (7x3 − x2)
Subtraction of Polynomials
You know that subtraction can be treated as adding the opposite. To find the opposite of a number, change the sign. Finding the opposite of a term just requires changing the sign of the coefficient. A polynomial is a sum of monomials, and each monomial is a term. To find the opposite of a monomial change the sign of its coefficient., Finding the opposite of a polynomial will mean changing all the signs. The opposite of 4t3 − 7t + 9t2 + 2 will be −4t3 + 7t − 9t2 − 2. Remember that the opposite is what you add to get to zero.
(4t3 − 7t + 9t2 + 2) + (–4t3 + 7t − 9t2 − 2) = (4t3 − 4t3) + (–7t + 7t) + (+9t2 − 9t2) + (2 − 2) = 0
Forming the opposite of a polynomial by changing the sign of each of its monomials is equivalent to multiplying the polynomial by -1. The opposite of 3x6 + 9x4 − 7x3 + 2x2 − 1 is − (3x6 + 9x4 − 7x3 + 2x2 − 1), and according to the distributive property, that equals (−1)(3x6) + (−1)(9x4) + (−1)(−7x3) + (−1)(2x2) + (−1)(−1) = −3x6 −9x4 + 7x3 − 2x2 + 1. This is why we sometimes use the expression “distribute the negative.” It’s actually a -1, but the effect is to change all the signs.
To subtract polynomials, you add the opposite of the second polynomial to the first polynomial. To perform the subtraction below, first rewrite it as adding the opposite.
(4x3 − 8x2 + 3x − 9) − (6x3 + 2x2 − 7x + 6)
(4x3 − 8x2 + 3x − 9) + (– 6x3 − 2x2 + 7x − 6)
Once you’ve changed to addition of the opposite, you can just go ahead and add as you’ve done before, either on a line or in vertical format. Here’s this example on a line.
(4x3 − 8x2 + 3x − 9) − (6x3 + 2x2 − 7x + 6)
(4x3 − 8x2 + 3x − 9) + (– 6x3 − 2x2 + 7x − 6)
(4x3 − 6x3) + (– 8x2 − 2x2) + (3x + 7x) + (−9 − 6)
−2x3 − 10x2 + 10x − 15
Let’s do the subtraction (y5 + 4y3 − 9y2 + 3) − (8y4 − 2y3 + 5y − 7) in vertical format. It may show another reason that adding the opposite is the best course of action. If you think of it as a subtraction problem, you face an awkward question once you line up like terms: what do you take 5y away from? There’s no term there.
The actual answer is that you’re subtracting 0 − 5y, but switching to adding the opposite removes the question. If you change to the opposite of the bottom polynomial, you can just add down each column.
When subtracting, it’s easy to make sign errors. Take the time to change all the signs of the bottom polynomial and then add. If you try to change signs as you go, it’s easy to change one or two and then forget to change the later ones.
Perform each subtraction by adding the opposite of the second polynomial.
21. (8x − 3) − (4 − 7x)
22. (9x2 − 7x + 5) − (– 2x2 + 6x + 3)
23. (2x2 − 3x) − (4x2 + 5x − 2)
24. (2y2 + 7y − 4) − (3y2 − 5 y + 2)
25. (5t2 − 7t + 2) − (3t2 − 9t − 1)
26. (2a2 + 6 − 3a) − (8 + 9a − a2)
27. (5x2 − 2x + 3) − (2x2 − x + 5)
28. (y3 + 5) − (y2 − 3)
29. (4x3 + 5x2 − 7) − (2x2 − 8x − 1)
30. (2x2 + 3x) − (5x2 + 2)
For longer problems that combine addition and subtraction, you don’t need any new procedures, but it is especially important for problems like these that you change subtraction to addition of the opposite. Trying to shift operations as you’re working will just set you up to make mistakes. Get everything over to addition, and then add all the way through. Let’s look at a couple of examples.
(3x − 7) − (x2 + 4) + (5x2 + 2x − 3) − (x2 − 2x– 1)
Change each subtraction to addition of the opposite.
(3x − 7) + (−x2 − 4) + (5x2 + 2x − 3) + (−x2 + 2x + 1)
Rearrange to group like terms.
(−x2 + 5x2 − x2) + (3x + 2x + 2x) + (− 7 − 4 − 3 + 1) = 3x2 + 7x − 13
One of the advantages of changing to adding the opposite is that it allows you to use a vertical alignment. You can add several polynomials in vertical arrangement but you can’t do that if you have some addition and some subtraction. Here’s an example.
(x2 + 7x − 3) − (2x2 − 3x − 1) + (5x2 + 8x) − (2x − 7)
Change to adding the opposite.
(x2 + 7x − 3) + (−2x2 + 3x + 1) + (5x2 + 8x) + (−2x + 7)
Now you can stack in vertical format.
Combine the polynomials by adding or subtracting as indicated. Give your answer as a polynomial in standard form.
31. (8x − 3) + (4 − 7x) − (4x − 7) + (9 − 2x)
32. (y3 + 5) + (y2 − 3) + (3y + y2) − (5y − 6y2)
33. (2x2 − 3x) − (3x2 − 2x − 5) − (4x2 + 5x− 2) + (−5x2 − 4x − 6)
34. (x2 − 3x + 4) − (− 7x2 − 5x) − (2x2 − 3x) + (4x2 + 5x − 2)
35. (3x2 + 2x − 1) + (5x2 − 7x + 6) − (7x2 − 15x − 3)
36. (2x2 + 3x) − (5x2 + 2) − (x2 − 8) + (5x2 + 6x − 1)
37. (8a3 − 4a2 + 6a + 2) − (2a2 + 6 − 3a) + (8 + 9a − a2)
38. (4x3 + 5x2 − 7) − (7x3 − x2) − (2x2 − 8x − 1) + (2x + 3x2) + (−5x3 − 4x)
39. (3t2 − 9t − 1) − (3t2 − 2t + 6) + (4− 6t2 + t)
40. (2x2 + 3x) − (−5x2 − 4x − 6) + (x2 − 2x − 5) + (4x2 − 5x − 2)
The Least You Need to Know
· Add polynomials by combining like terms.
· Use vertical alignment to organize like terms.
· Form the opposite of a polynomial by changing the sign of every term.
· To subtract a polynomial, add the opposite.
· For combined operations, change all subtractions to adding the opposite, and then add.