Polynomial Study Guide

Study Guide List Homework Pages

2/21 2/22 2/26 2/28

571­572/ 15­23, 45­52 578­579/ 24­29, 44­48, 50­58 586/ 35­53 599/ 12­35 606­607/ 20­54 (evens) 606­607/ 19­53 (odds) 613­614/ 18­57 (multiples of 3) 620­621/ 23­38 621/ 45­58

1. 2. 3. 4. 5. 6. 7. 8.

Adding and Subtracting Polynomials Multiplying polynomials Special Products of Polynomials Factoring x2 + bx + c Factoring ax2 + bx + c Factoring Special Products Factoring Cubic Polynomials Sum/Difference of two cubes

2/5 2/7 2/8 2/11 2/12

1. Adding and Subtracting Polynomials Adding Polynomials Adding polynomials is just a matter of combining like terms, with some order of operations considerations thrown in. As long as you're careful with the minus signs, and don't confuse addition and multiplication, you should do fine. You can add polynomials by grouping like terms and then simplifying. •

Simplify (2x

+ 5y) + (3x – 2y)

Clear the parentheses, group like terms, and simplify:

(2x + 5y) + (3x – 2y) = 2x + 5y + 3x – 2y = 2x + 3x + 5y – 2y = 5x + 3y The format you use, horizontal or vertical, is a matter of taste (unless the instructions explicitly tell you otherwise). Given a choice, you should use whichever format that you're more comfortable with. Simplify (3x3

+ 3x2 – 4x + 5) + (x3 – 2x2 + x – 4)

Horizontally:

(3x3 + 3x2 – 4x + 5) + (x3 – 2x2 + x – 4) = 3x3 + 3x2 – 4x + 5 + x3 – 2x2 + x – 4 = 3x3 + x3 + 3x2 – 2x2 – 4x + x + 5 – 4 = 4x3 + 1x2 – 3x + 1 Vertically:

Copyright © Elizabeth Stapel 2000-2007 All Rights Reserved

Either way, I get the same answer: 4x3

+ 1x2 – 3x + 1.

Subtracting Polynomials Subtracting polynomials is quite similar to adding polynomials, but you have that pesky minus sign to deal with. Here are some examples, done both horizontally and vertically: •

Simplify (x3

+ 3x2 + 5x – 4) – (3x3 – 8x2 – 5x + 6)

The first thing I have to do is take that negative through the parentheses. Some students find it helpful to put a "1" in front of the parentheses, to help them keep track of the minus sign: Horizontally:

(x3 + 3x2 + 5x – 4) – (3x3 – 8x2 – 5x + 6) = (x3 + 3x2 + 5x – 4) – 1(3x3 – 8x2 – 5x + 6) = x3 + 3x2 + 5x – 4 – 3x3 + 8x2 + 5x – 6 = x3 – 3x3 + 3x2 + 8x2 + 5x + 5x – 4 – 6 = –2x3 + 11x2 + 10x –10 Vertically:

Copyright © Elizabeth Stapel 2000-2007 All Rights Reserved

In the horizontal case, you may have noticed that running the negative through the parentheses changed the sign on each term inside the parentheses. The shortcut here is to not bother writing in the subtaction sign or the parentheses; instead, I'll change all the signs in the second row (shown in red), and add down:

Either way, I get the answer: •

Simplify (6x3

–2x3 + 11x2 + 10x – 10

– 2x2 + 8x) – (4x3 – 11x + 10)

Horizontally:

(6x3 – 2x2 + 8x) – (4x3 – 11x + 10) = (6x3 – 2x2 + 8x) – 1(4x3 – 11x + 10) = 6x3 – 2x2 + 8x – 4x3 + 11x – 10 = 6x3 – 4x3 – 2x2 + 8x + 11x – 10 = 2x3 – 2x2 + 19x – 10 Vertically:

Write out the polynomials, leaving gaps as necessary:

...and change the signs in the second line, and then add:

Either way, I get the answer: 2x3

– 2x2 + 19x – 10

Here’s the link for more problems with adding and subtracting polynomials. It’s only a page, so it’s up to you to find more problems. http://www.kutasoftware.com/FreeWorksheets/Ad ding+Subtracting%20Polynomials.pdf

2. Multiplying Polynomials Simple Polynomial Multiplication There were two formats for adding and subtracting polynomials: "horizontal" and "vertical". You can use those same two formats for multiplying polynomials. The very simplest case for polynomial multiplication is the product of two one-term polynomials. For instance: •

Simplify (5x2)(–2x3) You've already done this type of multiplication when you were first learning about exponents and variables. Just apply the rules you already know:

(5x2)(–2x3) = –10x5 The next step up in complexity is a one-term polynomial times a multi-term polynomial. For example: •

Simplify –3x(4x2

– x + 10)

To do this, I have to distribute the –3x through the parentheses:

–3x(4x2 – x + 10) = –3x(4x2) – 3x(-x) – 3x(10) = –12x3 + 3x2 – 30x The next step up is a two-term polynomial times a two-term polynomial. This is the simplest of the "multi-term times multi-term" cases. There are actually three ways to do this. Since this is one of the most common polynomial multiplications that you will be doing, I'll spend a fair amount of time on this. •

Simplify (x

+ 3)(x + 2)

The first way I can do this is "horizontally", where I distribute; in this case, however, I'll have to distribute twice: Copyright © Elizabeth Stapel 2000-2007 All Rights Reserved

(x + 3)(x + 2) = (x + 3)(x) + (x+3)(2) = x(x) + 3(x) + x(2) + 3(2) = x2 + 3x + 2x + 6 = x2 + 5x + 6

This is probably the most difficult way to do this multiplication. The "vertical" method is much simpler. First, think back to when you were first learning about multiplication. When you did small numbers, it was simplest to work horizontally, as I did in the first couple polynomial examples above:

3 × 4 = 12 But when you got to larger numbers, you stacked the numbers vertically and, working from right to left, took one digit at a time from the lower number and multiplied it, right to left, across the top number. For each digit in the lower number, you formed a row underneath, stepping the rows off to the left as you worked from digit to digit. Then you added. For instance, you would probably not want to try to multiply 121 by 32 horizontally, but it's easy when you do it vertically:

You can multiply polynomials in this same manner, so here's the same problem, but done "vertically": •

Simplify (x

+ 3)(x + 2)

Be sure to do your work very neatly. Set up the multiplication:

...and multiply:

So we get the answer:

x2 + 5x + 6

FOIL": A Special (and Misleading) Case There is also a special method, useful ONLY for a two-term polynomial times another two-term polynomial. The method is called "FOIL". The letters F-O-I-L come from the words "first", "outer", "inner", "last", and are a memory device for helping you remember how to multiply horizontally, without having to write out the distribution like I did, and without dropping any terms. Here is what FOIL stands for:

That is, FOIL tells you to multiply the first terms in each of the parentheses, then multiply the two terms that are on the "outside" (furthest from each other), then the two terms that are on the "inside" (closest to each other), and then the last terms in each of the parentheses. In other words, using the previous example: •

Use FOIL to simplify (x

+ 3)(x + 2)

"first": (x)(x) = x2 "outer": (x)(2) = 2x "inner": (3)(x) = 3x "last": (3)(2) = 6 So:

Copyright © Elizabeth Stapel 2000-2007 All Rights Reserved

(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6 •

Simplify (x

– 4)(x – 3)

So the answer is: x2 Using FOIL would give:

– 7x + 12

"first": (x)(x) = x2 "outer": (x)(–3) = –3x "inner": (–4)(x) = –4x "last": (–4)(–3) = +12 product: (x2) •

Simplify (x

+ (–3x) + (–4x) + (+12) = x2 – 7x + 12

– 3y)(x + y)

So the answer is: x2

– 2xy – 3y2

Using FOIL would give: "first": (x)(x) = x2 "outer": (x)(y) = xy "inner": (–3y)(x) = –3xy "last": (–3y)(y) = –3y2 product: (x2)

+ (xy) + (–3xy) + (–3y2) = x2 – 2xy – 3y2

"FOIL" works only for the specific and special case of a two-term expression times another twoterm expression. It does not apply in any other case. You should not rely on FOIL for general multiplication, and should not expect it to "work" for every multiplication, or even for most multiplications. If you only learn FOIL, you will not have learned all you need to know, and this will cause you problems later on down the road.

Here’s the link forproblems with multiplying special case polynomials. It’s only a page, so it’s up to you to find more problems.

http://www.kutasoftware.com/FreeWorksheets/Mul tiplying%20Special%20Cases.pdf

Here’s the link for problems with factoring special case polynomials. It’s only a page, so it’s up to you to find more problems. http://www.kutasoftware.com/FreeWorksheets/Mul tiplying%20Special%20Cases.pdf

Here’s the link for problems with factoring special case polynomials. It’s only a page, so it’s up to you to find more problems. http://www.kutasoftware.com/FreeWorksheets/Fac toring%20Special%20Cases.pdf

Here’s the link for problems with factoring by grouping polynomials. It’s only a page, so it’s up to you to find more problems. http://www.kutasoftware.com/FreeWorksheets/Fac toring%20By%20Grouping.pdf

if you’re looking for problems in the future, http://www.kutasoftware.com/ is an excellent website for free algebra worksheets. If you’re willing to spend $100, then u can download the kuta software for algebra 1. It’s the best software out there for algebra in my opinion. I use it and its definitely worth it.