Lesson 99-1: Adding and Subtracting Polynomials This is going to be a lot of definitions so pay attention and keep up. We all know what numbers are (6 or –5). We all know what variables are (x or a or q) We’ve all seen numbers and variables crammed together together (3x or –9y). We’ve even seen numbers and variables crammed together with exponents (2x3 or –3x5). You might even have seen more than one variable in there (7x2y3). All of these are examples of monomials. monomials. The number in front of a monomial is called the coefficient.. the coefficient The degree of a monomial is simply the total of all the exponents attached to variables. Usually, there’s only one variable, so it’s easy. The degree of 3x2 is 2. The degree of –7y5 is 5 Just add up those exponents. The degree of 2x2y9 is 11. exponents. 3 2 2 The degree of 2 a b is 4. (Be sure you don’t count it if it’s not attached to a variable.) The degree of 10 is 0. You could write that 10 . 1. Recall that x0 = 1, so it’s also 10x0. If the degree of a monomial is less than 0, then it’s not technically a monomial. (It’s a rational expression. We’ll deal with those another day.) A bunch of monomials added together is called a polynomial. polynomial. 2 2 2 3 4 3x + 2x – 5 3xy – 7x y – 2x + 4y The degree of a polynomial is the degree of its highest highest monomial. They’re properly written in order of descending degree. Ex. 1: What’s the degree of the two polynomials above? Are they written properly?
Polynomials with two terms are called binomials. binomials. Polynomials with three terms are called trinomials trinomials. ls. Polynomials with four terms are called…. polynomials. (Guess they got sick of naming them.) them.) Terms in polynomials that have exactly the same variables with exactly the same degree can be added together. Those that don’t… can’t. Ex. 2: Simplify: a) (2c2 – 8) + ((-3x2 + 4c – 1)
b) –2x2 + 3x – x3 + 3x2 + x3 – 12
c) 4x3 – 2x2 – 4 + x3 – 3x2 + x
d) (5y2 + 2y – 4) – ((-y2 + 4y – 3)
Lesson 99-2: Multiplying Polynomials Adding and subtracting polynomials is pretty straightforward. Multiplying them takes a little extra work. You should recall how to distribute: 2(x + 7) = 2x + 2(7) = 2x + 14 You can do the same thing with a monomial and a polynomial. polynomial. Ex. 1: Simplify: 3x2(2x3 – x2 + 4x – 3)
When you’ you’re multiplying two binomials, you you often hear it referred to as FOIL (2x + 3)(4x – 2) First multiply the First ones. Then multiply the Outer ones. Then multiply the Inner ones. Then multiply the Last ones. Ex. 2: 2: FOIL these: (3x – 2)( 2)(-2x + 5)
Multiplying polynomials together? Well, that’ll that’ll take longer, but it’s it’s not necessarily any more complicated. Suppose you have a binomial multiplied by a polynomial: (a + b)(c + d + e + f) Take the first element of the binomial and distribute it. a(c + d + e + f) = ac + ad + ae + af Then take the second element and distribute that. b(c + d + e + f) = bc + bd + be + bf Then glue them together ac + ad + ae + af + bc + bd + be + bf Let’s Let’s see a simpler simpler example. example. Ex. 3: Simplify (x + 4)(2x2 + 3x – 1)
If you want, you can treat it almost the same way you multiply numbers with multiple digits: Ex. 4: Simplify (3x2 – 2x + 1)(4x – 2)
3x2 - 2x + 1 X 4x - 2
Lesson 99-3: Special Polynomial Products Certain polynomial products show up repeatedly, so it’s it’s not a bad thing to remember them. Squared Binomials If you multiply multiply two identical binomials, that’s that’s just squaring them. 2 E.g.: (x + 5)(x + 5) = (x + 5) They always end up working the same way. Here’s Here’s the algebra: (a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2 Ex. 1: Expand: a) (x + 5)2
b) (3x – 2)2
The Sum and Difference Pattern You’ll You’ll regularly multiply two two binomials that are identical except for a sign switch. E.g.: (x + 5)(x – 5) This always works out the same, too. Here’ Here’s the algebra: (a + b)(a – b) = a2 – ab + ab – b2 = a2 – b2 The result here is called the difference of two squares. squares. Ex. 2: Expand: a) (x + 5)(x – 5)
b) ((3x 3x2 + 1)(3x2 – 1)
These patterns can occasionally be used to simplify certain seemingly more complicated situations. Ex. 3: Use special patterns to solve: a) 192
b) 18 . 22
Lesson 99-4: Solving Simpler Polynomial Equations Let us suppose that you have two quantities multiplied together. Let us call them A and B: A.B If A is 0, then it’s all 0. If B is 0, then it’s all 0. Therefore, if A . B = 0, then you know that at least ONE of those things has to be 0. Maybe both. Maybe just one. Not sure yet. Now suppose you have more specific quantities multiplied together and they together together equal 0. For example: x(x+1) = 0 Given what we just said before about A and B, one of those things has to be 0. (It can’t be both, obviously.) The question is: what values of x will make this equation true? It will be true if one of those those things is 0. That is, if x is 0 or if (x + 1) is 0. So break it up into two equations: x=0
x+1=0 x = -1
Those are your two solutions, the two ways that equation could be true. Ex. 1: Solve: (x – 1)(x + 5) = 0
Of course, you’ you’re not always going to see binomials binomials nicely separated out for you. you. You’ll You’ll likely need to take a polynomial and break it down into the smallest pieces you can. This is called factoring. factoring. For example, if you have 2x(x + 1) you can distribute to make it 2x2 + 2x Undoing that distribution process is the simplest way of factoring. factoring. Always start a factoring process by undoing distribution if you can. 1. Put parentheses around your polynomial. 2. Look at the coefficients. coefficients. Divide Divide out the largest number you can think of that goes into each coefficient evenly. Place that number out front of the parentheses. parentheses. 3. Keep doing #2 until you can’t can’t think of any more numbers you can pull out. Multiply those numbers together. 4. Now look at the variables. Divide out as many variables variables as are common to each term and put those variables out front with the number. Ex. 2: Factor 3x2 + 9x
Now something slightly more tricky: Ex. 3: 140 140x x4 – 70 70x x3 + 35x2
Now let’s let’s put it together: Ex. 4: Solve 2x2 + 6x = 0
And one more with a twist: Ex. 5: Solve 5x3 = 25x2
Remember:
WHEN SOLVING EQUATIONS EQUATIONS, DON’T EVER, EVER DIVIDE DIVIDE OUT A VARIABLE!
Lesson 99-5: Factoring x2 + bx + c You know that (x + 4)(x + 3) can be FOILed. You’ll get x2 + 7x + 12. Well, how do you undo that? How do you factor that second thing to get the first one back? What are those two numbers? Easy as pie. (Mmmmm… pie…..) Look carefully at that 12. Is it not interesting that 12 = 3 . 4? And take a look at that 7. Is it not interesting that 7 = 3 + 4? (The correct answer to both questions is “Yes.’) If you have something that looks like x2 + bx + c, all you need is to find two things that multiply together together to make c but add together to make b. FOIL it back out if you want to check. Ex. 1: Factor the following: a) x2 + 7x + 10
b) x2 + 10x + 24
Seems simple enough. It gets slightly more difficult when negatives show up, but the process actually doesn’t change at all. Ex. 2: Factor Factor the following: 2 a) x + 3x – 18
b) x2 – x – 20
c) x2 – 5x + 6
d) –x2 – 5x + 36
A table you might find useful: When factoring x2 + bx + c… c b You’re looking for… + + 2 positive numbers + - 2 negative numbers - + 1 negative, negative, 1 positive number (positive is bigger) - - 1 negative, 1 positive number (negative is bigger) So now you can use this kind of factoring to solve simple trinomials: Ex. 3: Solve: a) x2 + 5x – 14 = 0
b) x2 = 2x + 15
Lesson 99-6: Factoring ax2 + bx + c All I did was add an “a” up there, and it suddenly got harder. If you have (2x + 1)(3x + 4), you can FOIL it. It becomes 6x2 + 11x + 4. Going backwards is still factoring, but now it’s it’s just plain harder. There isn’t isn’t a simple trick here. You just have to wrestle with it, but there is a process. To factor something that looks like ax2 + bx + c 1. Set up two sets of parentheses to hold binomials: ( )( ) 2. Drop 2 x’ ( x )( x ) x’s at the front 3. Find two numbers that multiply together to make a; place them in front of the x’s. x’s. 4. Find two numbers that multiply together to make c; place them at the back of the parentheses, adding adding pluses where necessary to make binomials binomials 5. Test to see if your guess is correct by FOILing it out. a. If it FOILs correctly, correctly, stop. You finished factoring. b. If you don’t don’t get your original trinomial, erase, go back to step 3 or 4, 4, and try again. The good news is that the table from the previous lesson is still true, so I’ll stick it here again for handy reference. When factoring x2 + bx + c… c b You’re looking for… + + 2 positive numbers + - 2 negative numbers - + 1 negative, 1 positive positive number (positive is bigger) - - 1 negative, 1 positive number (negative is bigger) Ex. 1: Factor 2x2 – 13x + 6
Ex. 2: Factor 4x2 + 11x – 3
Ex. 3: Factor –3x2 – 13x – 4
It’s It’s been a while, but remember that the first thing you do in ANY factoring process is to check to see if you can undo a distribution. Can you factor out out a common value?
Ex. 4: Solve: -16t2 + 46t + 6 = 0
Lesson 99-7: Factoring Special Products Back a few lessons, we noted there were special products. This is just designed to help you recognize and undo those. those. Difference of two squares Recall that (a + b)(a – b) = a2 – b2 Any time you see two squares with a minus in between, you can factor them easily. Take the square root of each and drop them into binomials like we started with. Ex. 1: Factor: a) x2 – 100
b) 4x2 – 81
c) x2 – 49y4
d) 12 – 48x2
Perfect Square Trinomials This one’ one’s a little trickier to spot. In fact, you might not spot it until after you’ve you’ve factored it like we did in the last two lessons. And if that happens, happens, it’s it’ll save you a little time. it’s perfectly okay. But if you do spot it, then it’ll 2 2 2 Recall that (a + b) = a + 2ab + b If you have a trinomial: • Is the first term a perfect square? • Is the last term a perfect square? • If so, check to see if you can multiply the two square roots together and then multiply by two to get the middle term. • If that works, drop the square roots into a binomial and square it. The factoring is over.
Ex. 2: Factor a) x2 + 6x + 9
b) 4n2 + 20n + 25
c) 9c2 – 6cd 6cd + d2
d) –2x2 – 16x – 32
Now you can use this to solve equations, including ones that are kind of a pain in the rear, like this one: 2 Ex. 3: Solve: x − 5 x +
25 = 0 (There are two ways to solve here… here…) 4
Lesson 99-8: Final Thoughts on Factoring You’ve You’ve factored out numbers. You’ You’ve factored out variables. Now, let’s let’s factor out binomials. Hey, anything that’s that’s common, you can yank that out. Ex. 1: Factor: Factor: a) 4x(x – 3) 3) + 5(x – 3)
b) 2y2(y – 5) – 3(5 3(5 – y)
Occasionally (VERY occasionally… occasionally…) this technique can be used to simplify polynomials of 4 terms. terms. • Split the polynomial into 2 groups • Factor from each group • If the binomials you created are the same, factor the binomials Ex. 2: Factor: a) x3 + 2x2 + 8x + 16
b) a2 + 4a + ax + 4x
It is often handy to factor everything you can to get things into their most basic components. This is called factoring completely. Here are some reminders of what you can and should try to do: • • • •
Factor Factor Factor Factor
out individual numbers and variables (un(un-distribute) the difference of two squares or a perfect square trinomial a trinomial like ax2 + bx + c into two binomials by grouping like we did above
For a polynomial to be factored completely, you shouldn’t shouldn’t be able to do any of those things anymore. You’ You’ve done them all you can. Ex. 3: Factor completely: a) x2 + 2x – 1
b) 3x3 – 21x2 – 54
c) 8x3 + 24x
Use these factoring techniques to solve polynomial equations. Ex. 4: Solve 2x3 – 18x2 = -36x
