Transcript Power Point Version

MTH 209
The University of Phoenix
Chapter 5
Factoring Polynomials
Operations with Rational Expressions
Chapter 5 Section 1
The Basics of Factors
• Just the Fact-or Ma’am.
• or
• The O’Righty Factor
• (I’m getting the puns out of my system).
Primes are your friend…
• A prime number is a rock bottom, hard end, stop playing
with the math, number.
You can ONLY divide it by 1 and itself.
• 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61
67 71 73 79 83 89 97 101 103 107 109 113 127 131
137 139 149 151 157 163 167 173 179 181 191 193
197 199 211 223 227 229 233 239 241 251 257 263
269 271 277 281 283 293 307 311 313 317 331 337
347 349 353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463 467 479
487 491 499 503 509 521 523 541 etc.
http://www.utm.edu/research/primes/lists/small/1000.txt
Prime Factorization of IntegersPulling out the Primes
(and puttin’ on the Ritz)
12 = 2 · 6
12 = 1 · 12
12 = 2 · 2 · 3 which is 22 · 3
A Prime Number is a positive integer larger
than 1 that has NO integral factors other
than itself and one.
Beyond Primes?
• All the REST of the numbers that are
(basically) made of prime numbers
multiplied together (and can be therefore
factored) are called composite numbers.
Example 1 page 320
Prime Factorization
• We write the number as two integers (anything
you can ‘see’) .
• 36 = 2 · 18
then break up the next thing you
can
• = 2 · 2 · 9 and keep breaking composite numbers
apart
• =2 · 2 · 3 · 3 and we are almost done…
• = 22 · 32 then covert them to exponents if
possible ** Ex 7-12**
Le Example’ 2 page 321
• Find the prime factorization for 420
• We start by dividing by the smallest prime
number we can see will go into it an integer
7
number of times
5 35
• How about 2?
3 105
• Start Here 
** Ex 13-18**
2
2
210
420
And the winner is…
• So the answer is 2 · 2 · 3 · 5 · 7 or (one
more step) 22 · 3 · 5 · 7
It’s only in terms of prime numbers!
Side note: For that first step, if the number is
even, divide by 2, if it odd try 3. If it ends
in a zero or 5, start with 5.
The GREATEST common factor
• This adds to your toolkit when working
with polynomials soon…
• We want to know what the biggest number
two numbers share as a factor.
It’s called the GCF by friends
• Here’s how it works:
8 = 2 · 2 · 2 = 23
12= 2 · 2 · 3 = 22 · 3
When you expand it, you see, magically, that
there are two 2’s in each.
8 = 2 · 2 · 2 = 23
12= 2 · 2 · 3 = 22 · 3
The cookbook
• Following what we just did for those two
numbers:
• 1) Find the prime factorization of each
integer (break it down all the way)
• 2) Determine which primes are shared by
both (including multiples like 22).
• 3) Multiply the shared primes together and
Voil’a!
Special Wonderfullness
• If there are NO common factors, then the
only common number is 1.
• Nice when THAT happens… no?
Example 3 page 322
• a) 150, 225
5
5 25
3
2
5
5 25
3
75
150
So 150= 2·3 · 52 and 225 = 32 · 52
The GCF is 3 · 52 = 75
3
75
225
ex 3b
• The numbers 216, 360, 504
• Using the same long division like trick, you can
find
• 216=23 · 33 · 3 and 360=23 · 32 · 5 and 504=23 · 32 · 7
• 216=23 · 32 · 3 and 360=23 · 32 · 5 and 504=23 · 32 · 7
• The answer is 23*32 = 72
3c
• Now for 55 and 168
• 55= 5 · 11
168=23 · 3 · 7
• There are NO primes in common, so the
GCF = 1.
** Ex 19-28**
Example 4a page 323
On the road to polynomials, one
must start with monomials
• What’s the GCF of 15x2 and 9x3
• 15x2 = 5 · 3 · x · x
while 9x3 = 3 · 3 · x·x·x
• They share 3·x·x so the GCF is 3x2
** Ex 29-40**
The cookbook for monomials…
• 1) Find the GCF for the coefficients of the
monomials
• 2) Form the GCF from the GCF of the
coefficients, then the variables in the same
manner.
pt b
• 12x2y2
30x2yz
42x3y
12x2y2= 2·2·3·x·x·y·y
30x2yz = 2·3·5·x·x·y·z
42x3y= 2·3·7·x·x·x·y
 6x2y
** Ex 29-40**
The why behind the madness
• When you have a binomial you need to
factor, you need to take out the biggest
chunks you can. The GCF is the biggest
chunk you can take!
Example 5 pg 323
GCF’s and Polynomials
• a) 25a2+40a
The GCF from 25 and 40 is 5.
Pull out the 5.
And we can pull out an ‘a’.
5a(5a+8)
5b
• 6x4-12x3+3x2
• We can pull out a 3. That’s the best we can do!
• Then we can take out an x2 as well…
• 3x2(2x2-4x+1)
5c
• x2y5+x6y3
• You can take out an x2 and a y3 from both.
You don’t have any of those nasty
coefficients to deal with.
• x2 y3(y2+x4)
• ** Ex 41-68**
Ex 6a page 324
A binomial factor
• (a+b)w+(a+b)6
• Why not take off an (a+b) from both terms?
• (a+b) (w+6)
Ex 6b
• x(x+2)+3(x+2)
• Both have an (x+2) in them, so pull it out
• (x+2)(x+3)
Ex 3c
• y(y-3)-(y-3)
• Both have a (y-3) in them!
Take it to the left
(y-3)(y-1)
** Ex 69-76**
The common danger…
If you take the exact term out when factoring,
you MUST leave a “1” in it’s place.
ab+b take out the b! b(a+1) NOT b(a)
Multiply it out to check!
Worrying about the Opposite of
the GCF
• Sometimes it’s best to take out the – of the
GCF if it makes the end answer look better.
• Starting with –4x+2xy you could take out
the 2x and get 2x(-2+y)
• You might also take out –2x and get
-2x(2-y) You are forcing the y to become
negative.
Let’s see it in action in Ex 7
pg 325
• We’ll do it BOTH ways…
a) 3x-3y  3(x-y)
or  -3(-x+y)
b) a-b  1(a-b) sure you can factor out a 1
or  -1 (-a+b)
c) -x3+2x2-8x  x(-x2+2x-8)
or  -x(x2-2x+8) ** Ex 77-92**
Danger!
• Be sure you change the sign of EVERY
term in the ( )’s when you pull out that
negative sign!
Doing the Factor
Factoring the Doing
•
•
•
•
•
•
•
Definitions Q1-Q6
Find the prime factors Q7-18
Find the GCF integers Q19-Q28
Find the GCF monomials Q29-40
Factor out the GCF in each expression Q41-76
Factor out the GCF then the opposite Q77-92
Word like problems Q 93-98
Section 5.2 Back to the Future
The ‘Special Factors’
• Last time we looked at the special factors
like (a+b)2 and (a-b)2 and (a+b)(a-b)
• Last things first… well take (a+b)(a-b)
• Remember : (a+b)(a-b) = a2-b2
Example 1 page 329
• a) y2-81 First, note the square root of 81 is
9, so you’re going to have 9 as the number
in the factored polynomials.
• So this is really y2-92 (right?)
• Finally that comes from (y+9)(y-9)
• done!
Ex 1b
• 9m2=16 Next, move 16 over to the left…
• So we have something familiar: 9m2-16=0
• Again, the number by itself is 16 which is
42.
• Our squared variable has a 9 in front of it,
and that is just 32
• This gives us 32m2-42 = (3m)2-42=
(3m+4)(3m-4)
Example 1c
• 4x2-9y2
Note 4x2=(2x)2 and 9y2=(3y)2
• So  (2x+3y)(2x-3y)
** Ex 7-20**
On to factoring Perfect Square
Trinomials
• Note we are just doing the polynomial
section in reverse.
In 4.6 we saw…
• (a+b)2= a2+2ab+b2
• So we saw:
x2+6x+9 = x2 + 2 x 3 + 32
which is
a2 + 2 a b + b 2
We’ll be going backwards now we know what
a and b are and get  (x+3)2
The trick to expose them…
• Is it a perfect square trinomial?
• 1) The first and last terms are a2 and b2
• 2) The middle term is 2ab or –2ab
( So, to see it quickly, take half of the middle
coefficient and see if that squared is the
b2ed term. )
Example 2 page 330
Identifying special products…
• a) x2-14x+49 Let’s try x as the first term and ½(14) for
the second. That is 7 and 72 IS 49. It works (So the
middle term is –2 · x · 7). This is a perfect square
trinomial
• b) 4x2-81 Both terms are perfect squares (2x)2 and 92. So
this is the difference of two squares.
• c) 4a2+24a+25 The first term is (2a)2 and the last term is
(5)2 . The middle term would be 2·2a·5 =20a BUT it’s
really 24a so this is NOT a perfect square trinomial.
• d) 9y2-24y-16 The first term is (3y)2 and the last term is
(4)2. Our middle term would be 2 · 3y · 4 = 24y2 . But in
a perfect square trinomial, the first and LAST terms are
always positive. It’s not true and not one.
• ** Ex 21-32**
Actually Factoring them…
• Remember:
• a2+2ab+b2 = (a+b)2
• a2 -2ab+b2 = (a-b)2
• Index cards…
Ex 3 page 331
• a) x2-4x+4 The first term is (x)2 and the
last term is 22 . The middle term should be
-2·x·2 or –4x. True!
• So this is  (x-2)2
• b) a2 +16a+64 The first term is (a)2 while
the last term is 82 . The middle should be
2·a·8 =16a True!
• So this is  (a+8)2
** Ex 33-50**
Ex 3c
• c) 4x2-12x+9
• The first term is (2x)2 and the last is 32 .
• The middle term is negative, does this
work? -2 ·2x·3 = -12x Yes!
• (2x-3)2
** Ex 33-50**
The end of the road in factoring
• Factoring Completely has happened when
you can’t go any further.
• If you can’t factor a polynomial it is called a
prime or irreducible polynomial.
• For example: 3x or w+1 or 4m-5
• When your factored polynomial is only
made up of prime polynomials, you’re
DONE!
Example 4 page 331
• a) Factor each completely:
2x3-50x = 2x(x2-25) = 2x(x+5)(x-5)
b) 8x2y-32xy+32y = 8y(x2-4x+4) = 8y(x-2)2
** Ex 51-70**
Safety in Groups (or numbers)
•
•
•
•
Factoring by Grouping
remember when we had (x+a)(x+3)?
If you multiply these out, you get:
(x+a)x+(x+a)3 THIS is the important form
for us, since it is the first step.
• x2+ax+3x+3a messy! THIS is what your
problems will usually look like.
Grouping Pt 2
•
•
•
•
•
So if you are given a big long mess like
w2-bw+3w-3b
(w2-bw)+(3w-3b) The GROUP in group
w(w-b)+3(w-b) The factor in GROUPing
(w+3)(w-b) The second factoring in grouping
Example 5a Grouping
• xy+2y+3x+6 we look at it and see we can
pull a y out of the first two terms and a 3
out of the second two terms
• (xy+2y) + (3x+6)
• y(x+2)+3(x+2)
• (y+3)(x+2)
5b Grouping
•
•
•
•
2x3-3x2-2x+3
(2x3-3x2)+(-2x+3) just group
x2(2x-3)+1(-2x+3) pull out x2 and 1
What is in the ( )’s is almost the same,
except for –1 in the second term
• x2(2x-3)-1(2x-3) factor out –1 !
• (x2-1)(2x-3)  (x-1)(x+1)(2x-3) Done!
Ex 5c
• ax+3y-3x-ay
= ax-3x-ay+3y rearrange the terms
= (ax-3x)+(-ay+3y)
= x(a-3)-y(a-3)
= (x-y)(a-3)
** Ex 71-86**
Fun is Factoring Using Neurons
•
•
•
•
•
•
•
Definitions Q1-6
Factor polynomials Q7-20
What kind binomial is it? Q21-32
Factor perfect square trinomials Q33-50
Factor each polynomial Q51-70
Factor each polynomial completely Q71-98
Word problems Q89-94
Section 5.3 Sleuthing out
ax2+bx+c where a=1
• This is the next step… beyond this (5.4)
we’ll see a ≠1 (but not yet…)

•
•
•
•
Going backwards first:
(x+2)(x+3) = (x+2)x+(x+2)3
x2+2x+3x+6
x2+5x+6 This is what the problems will
look like.
Example 1a pg 336
• a) x2+5x+6
• We’re looking for two numbers added that
make 5 and when multiplied together make 6
• x2+2x+3x+6
• (x2+2x)+(3x+6)
group again!
• x(x+2)+3(x+2)
• (x+3)(x+2)
Example 1b
•
•
•
•
•
x2+8x+12
x2+2x+6x+12
(x2+2x)+(6x+12)
(x+2)x+(x+2)6
(x+2)(x+6)
• You can use FOIL to check your answer!
Example 1c
• a2-9a+20
• What two numbers multiplied together make 20
but when added make –9?
• How about –4 and –5 ?
• a2-4a-5a+20
• (a2-4a)+(-5a+20)
• a(a-4)-5(a-4) I took out a –5 since I want the –4
inside to match the first a-4
• (a-5)(a-4)
** Ex 7-20**
Example 2 pg 337
Factoring the Illusive Trinomial
• a) x2+5x+4 I know 1+4=5 and 1·4=4
so I can just write down (x+4)(x+1)! done
b) y2+ 6y-16 I know that 8-2 = 6 and 8·-2=-16
so it must be (y+8)(y-2)
c) w2-5w-24 I’ll try –8 +3 = -5 and –8·3=-24
good so (w-8)(w+3)
** Ex 21-28**
Example 3 page 338
More of the same!
• a) 2x-8+x2 reorder
• x2 + 2x -8 and 4-2=2 and 4·-2 = -8
• (x+4)(x-2)
• b) -36+t2-9t reorder to : t2 -9t -36
and with some though –12+3 = -9 and –
12·3=-36 good so (t-12)(t+3)
** Ex 29-30**
Prime Polynomials – you can’t
get there from here Ex 4a pg 328
• a) x2+7x-6 are there any numbers that work?
That add to 7 and multiply to become –6?
Product
Sum
• -6=(-1)(6)
-1+6=5
• -6=(1)(-6)
1+(-6)=-5
• -6=(2)(-3)
2+(-3)=-1
• -6=(-2)(3)
-2+3=1 None work.
This IS done (prime) already.
Ex 4b
• x2+0x+9
•
•
•
•
Product
Sum
9=(3)(3)
3+3=6
9=(-3)(-3)
-3+(-3)=-5
9=(9)(1)
9+1=10
9=(-9)(-1)
-9+(-1)=-10
Again, none work.
This IS done (prime) already. ** Ex 31-58**
The quick answer to the
Sum of Two Squares
• a2+b2 has no common factor other than 1
• It is always a prime polynomial!
• Easy cheesy!
Yet another baby step, we’ll do
the same now but mix in another
variable Ex 5- Page 339
• a) x2+2xy –8y2
We can guess that 4 and –2 will take care of
our number parts (sum=2, product=-8)
To get the 2y and –8y2 we need to add a y to
to our 4 and –2 for 4y and –2y.
So writing it we see (x+4y)(x-2y)
Polynomials with two variables
• b) a2-7ab+10b2
• We’ll need to guess that –5 and –2 will
work (sum = -7 and product = 10)
• There is an extra b in the last term so we’ll
use -5b and –2b
• Writing it out (a-5b)(a-2b) ** Ex 59-66**
Just more factoring for the visuals…
Ex 6 – pg 340
•
•
•
•
a) x3-6x2-16x
factor out the GCF x
x(x2-6x-16) then deal with the polynomial
How about 2 and -8? Sum = -6, Product =-16
x(x-8)(x+2)
part b
•
•
•
•
•
b) 4x3+4x2+4x
factor out the GCF 4x
4x(x2+x+1) then deal with the polynomial
How about 1 and 0? Sum = 1, Product =0
How about 1 and 1? Sum 2 , Product = 1
it’s done… it’s a prime polynomial
(bonus) part c
•
•
•
•
•
3wy2+18wy+27w factor out the GCF 3w
3w(y2+6y+9) then deal with the polynomial
How about 3 and 3? Sum = 6, Product =9
3w(y+3)(y+3)
3w(y+3)2
** Ex 37-108**
Section 5.3 Trial and Trial
•
•
•
•
Definitions Q1-6
Factor each trinomial Q7-20
Factor (if possible) each polynomial Q21-108
Spoken problems with words in them Q109-114
Section 5.4 Factoring ax2+bx+c
with a  0
• The ac Method
• (not the ac/dc method – but rock on!)
• We really mean the a and c in ax2+bx+c
• Since a is some non-1 and non-0 number,
we want to find the product of ac that sums
to b. (We did this before, but a=1).
How it runs…
•
•
•
•
2x2+7x+6
ac=2·6=12
2x2+3x+4x+6 change 7x to 3x+4x 3·4=12
(2x+3)x+(2x+3)2 do the factor thing
(2x+3)(x+2) Factor our 2x+3
The Strategy
• To factor the trinomial ax2+bx+c
• 1. Find two numbers that have a product
equal to ac and a sum equal to b
• 2. Replace bx by two terms using the two
new numbers as coefficients
• 3. Factor the resulting four-term polynomial
by grouping.
Example 1 page 344
• a) 2x2 + x –6
2·(-6) = -12
Possible pairs 1 and –12, -1 and 12, 2 and –6, -2
and 6, 3 and –4, -4 and 3 Which works?
-3 and 4!
2x2 –3x + 4x –6 = (2x2-3x)+(4x-6) =
(2x-3)x+(2x-3)2 = (2x-3)(x+2)
Ex 1b
b) 2x2+x-6
2·(-6) = -12
1 and –12, 2 and –6, 3 and –4
-1 and 12, -2 and 6, -3 and 4
Change x to –3x and 4x
= 2x2+x-6 = 2x2 –3x+4x-6
=(2x-3)x+(2x-3)2
= (2x-3)(x+2)
Example 1c
• 10x2+13x-3  ac = -30
• What two numbers sum to 13 and have a
product of –30?
• 1 and –30, -1 and 30, 2 and –15, -2 and 15, 3
and –10, -3 and 10, 5 and –6, -5 and 6
• 10x2-2x+15x –3= (5x-1)2x+(5x-1)3=
• =(5x-1)(2x+3)
** Ex 5-42**
Example 2 pg 345
8x2-14xy+3y2
ac=24 we can look at:
-1 and –24, -3 and –8, -2 and –12, -4 and –6
Only –2 and –12 sum to –14
8x2-14xy+3y2=8x2-2xy-12xy+3y2=
(4x-y)2x+(4x-y)(-3y)=
(4x-y)(2x-3y)
** Ex 43-48**
Trial and Error
• 3x2+7x-6
• The factors of 3x2 can only be 3x and x (right?)
• The factors of 6 can only be 6,3,2,or 1
(3x 3)(x 2) or (3x 2)(x 3) or (3x 6)(x 1) or (3x 1)(x 6)
The negative “c” means we have a + and – in the factors
(3x + 3)(x - 2) This gives a middle term of –3x wrong
(3x - 3)(x + 2) This gives a middle term of 3x wrong
(3x + 2)(x - 3) And this gives a –7x
(3x - 2)(x + 3) While this gives 7x We GOT IT!
Example 3 page 346
Trial and Error
•
•
•
•
•
•
2x2+5x-3
Our pieces: 2x2 can only be 2x and x
1 and 3 are the only ones for 3
(2x 1)(x 3) and (2x 3)(x 1)
The negative sign means we have a + and –
(2x - 1)(x + 3) gives us 5x and the answer!
Ex 3b
•
•
•
•
•
3x2-11x+6
3x2 can only be 3x and x
6 can only be 6,1 and 2,3
(3x 1) (x 6) or (3x 2)(x 3) or (3x 6)(x 1) or (3x 3)(x 2)
And with the c positive and b negative, then both signs
must be negative
• Trying the first gives –18x wrong
• Trying the second gives –11x GREAT!
• (3x-2)(x-3) Good.
**49-68**
The cookbook
• 1. If the signs of the terms of a trinomial are + +
+, then both factors are sums.
• x2+5x+6 = (x+2)(x+3)
• 2. If the signs are + - + then both factors are
differences.
• x2-5x+6 = (x-2)(x-3)
• 3. If the signs are + + - or + - - then one factor is a
sum, the other a difference
• x2+x-6 = (x+3)(x-2) and x2-x-6=(x-3)(x+2)
Why not add another variable
into our trial and error method
• (Do we HAVE to Mom?)
Example 4
• Factor 6x2-7xy+2y2
• We can have
• (3x 2y)(2x y) or (3x y)(2x 2y) or
(6x 2y)(x y) or (6x y)(x 2y)
• Both must be negative… try a few in your
head… and bango… (3x-2y)(2x-y)
** Ex 69-72**
Example 5 page 348
Factoring Completely (Doing what it takes)
• Factor each polynomial completely:
a) 4x3+14x2+6x
– take out a 2x  2x(2x2+7x+3)
– Then expand the trinomial
– 2x(2x+1)(x+3)
Example 5b
•
•
•
•
12x2y+6xy+6y
Take out the 6y
6y(2x2+x+1)
Then use the ac method 2·1=2 and 1·2=2
but neither added = 1 so we are finished.
That’s a prime polynomial.
** Ex. 79-88**
Example 6 Cleaning up…
Factoring out the opposite of the
GCF
• Best when a is negative.
• a) –18x2+51x-15x
-3x(6x2-17x+5) = -3x(3x-1)(2x-5) the two numbers
that multiply to 5 and with the 3 and 2 in front 2x-15x = -17x
• b) -3a2+2a+21
-1(3a2-2a-21) = -1(3a+7)(a-3)
the two numbers that multiply to become –21 and
with the 3 and 1 in front add to 7a-9a = -2a.
** Ex. 89-104**
You MUST practice these to get
good! Section 5.4
•
•
•
•
•
•
•
•
Definitions Q1-4
Playing with the numbers Q5-10
Factor a trinomial using ac Q11-16
Factoring it out Q17-42
Complete the factoring Q43-48
Use trial and error Q49-72
Factor any polynomial thrown at ya Q73-104
Dialect Problems Q105-110
Section 5.5 Linking the Division
Stuff to Factoring: The Strategy
• If you are told to factor a big integer like
1001, you might guess or be told that a
factor is 77.
• So 1001 = 77 · 13
• Then you can break 77 up
• And 1001= 11·7·13
Example 1 page 352
You are given one prime factor
of a polynomial, find the rest…
• You are given x+1 is a prime factor of
x3+2x2-5x-6
x 2  x  6............................
x 3  2 x 2  5 x 2  6................
x 1 3
x  x 2 ...............................
x 2  5 x 2 .................
x 2  x...................
6 x  6...........
6 x  6...........
0
Ex 1 continued
• So we know (x+1) is one and the answer is
the rest…
• (x+1) (x2+x-6)
• (x+1)(x+3)(x-2) is it!
** Ex. 7-18**
Shortcuts
• starting with a3-b3 and a3+b3
And we are told that the first has (a-b) in it
and the second has (a+b) in it.
If we factor those primes out of the two
binomials we can do the division (do it once
and we don’t have to worry about it again).
Doing the work once…
2
2
a

ab

b
............................
a  ab  b ............................
3
2
2
2
3
2
2
2
a

0
a
b

0
ab

b
................
a  0a b  0ab  b ............. a  b
3
2
a b 3
2
a

a
b...............................
a  a b.................................
2
2
2
2

a
b

0
ab
............
a b  0ab ............
a 2b  ab 2 .............
a 2b  ab 2 .............
ab 2  b 2 ..........
ab 2  b 2 ..........
ab 2  b 2 ..........
ab 2  b 2 ..........
0
0
2
2
Factoring a Difference of Sum of
Two Cubes
a3-b3 = (a-b)(a2+ab+b2)
a3+b3 = (a+b)(a2-ab+b2)
Always…!
Example 2 page 353
For these, just use the overall rules above
and plug in the numbers… no more
thinking than that!
• a) w3-8 note 23=8 so a=w, b=2 and use the
difference formula
= (w-2)(w2+2w+4)
b) x3+1 note a=x and b=1 use the sum formula
= (x+1)(x2-x+1)
c) 8y3-27 note a=2y and b = 3 use the difference one
and get: (2y-3)(4y2+6y+9)
** Ex. 19-34**
Example 3 page 354
4th powers too much for us?
a) x4-16 = (x2)2 –42 = (x2-4)(x2+4)=
(x-2)(x+2)(x2+4)
b) 81m4-n4=(9m2)2-(n2)2= (9m2-n2)(9m2+n2)
(3m-n)(3m+n)(9m2+n2)
** Ex 35-42**
The overall winning Strategy for
completely factoring polynomials
1.
2.
3.
4.
5.
6.
If there are any common factors, take them out first.
Next see if it is a difference of two squares or cubes, or a
sum of two cubes. A sum of two squares does not factor.
When factoring a trinomial, check to see if it is a perfect
square
If it’s not a perfect square, then use the ac method or trial
and error
If the polynomial has four terms, try grouping.
Then check to see if any of the factors can be factored
again… and on…and on until all are primes.
Example 4 Potpourri
a) 2a2b-24ab+72b
=2b(a2-12a+36)
=2b(a-6)2
b) 3x3+6x2-75x-150 take out a 3
=3[x3+2x2-25x-50]
group and factor
=3[x2(x+2)-25(x+2)]
factor again
=3(x2-25)(x+2)
expand the perfect square
=3(x+5)(x-5)(x+2) done!
Example 4 Potpourri
c) –3x4-15x3+72x2 Take out –3x2
–3x2(x2+5x-24)
We want a product of –24 and a sum of 5
What numbers do that? 8 and –3
–3x2(x2+8x-3x-24) = –3x2(x-3)(x+8)
Example 4 Potpourri
c) 60y3-85y2-25y Take out 5y
5y(12y2-17y-5)
We want a product of -60 and a sum of -17
What numbers do that? –20 and 3
5y(12y2-20y+3y-5) = 5y[4y(3y-5)+1(3y-5)]=
5y(3y-5)(4y+1)
** Ex. 43-110**
Bonus Example
a) Factor each completely
ax3+ax
= ax(x2+1) and that is it!
b) by3+b
= b(y3+1) = b(y+1)(y2-y+1)
c) 8m3+22m2-6m
= 2m(4m2+11m-3) = 2m(4m-1)(m+3)
Uncle Sam wants YOU to factor.
Section 5.5
•
•
•
•
•
•
•
Definitions Q1-6
Given one prime, factor the polynomial Q7-18
Factor each difference or sum of cubes Q19-34
Factoring a Difference of two fourth powers Q35-42
Factor anything thrown at you Q36-110
A smattering of applications Q111-114
More practice Q115-150
Section 5.6
Enter the Quadratic Equation
• Don’t panic!
• It’s just a special polynomial!
• It looks like ax2+bx+c=0 no big deal!
(you’ve already worked with ax2+bx+c stuff)
• (Bailiff, bar the doors)
The Quadratic Equation
• ax2+bx+c=0
• If a,b, and c are real numbers and a isn’t 0
• This is called a quadratic equation
Behind the scenes…
• To solve these uses the idea of the
“Zero Factor Property”
• If the equation a * b = 0 then either a = 0,
b= 0 (or both, of course) .
You get to use all your factoring
tools!
• First you factor.
• Then you use the Zero Factor Property
• (ZFP?)
Example 1 pg 360
• Use the ZFP to solve x2+x-6=0
x2+x-6=0
(x+3)(x-2)=0
So either x+3=0 or x-2=0
So either x=-3 or x=2
Example 1 continued
x2+x-6=0
So either x=-3 or x=2 is the solution… plug them
in…
9-3-6=0 yes or 4+2-6= 0 yes
So BOTH are answers -3 and 2
** Ex. 7-10**
Example 2 pg 360
•
•
•
•
•
•
•
•
Solve 3x2=-3x
Coral EVERYTHING to the left…
3x2+3x=0
Factor : 3x(x+1)=0
Either 3x=0 or x+1 = 0
So either x=0 or x=-1
Check x=0  0=0 yes, x=-1 3-3=0, 0=0 yes
Both x=0 and x=-1 are solutions (This is called a
compound equation). ** Ex. 19-26**
Caution: don’t play with your
food
•
•
•
•
In the previous example 3x2=-3x
We DON’T divide by 3 or 3x before hand.
Dividing by 3x would make it x=-1.
We’d only get ONE of the two answers.
The Cookbook!
1. Rewrite the equation with all stuff to the left.
2. Factor the left hand side completely
3. Use the zero factor property to get simple linear
equations
4. Solve the linear equations
5. Check the answers
6. State the solution(s) to the original equation.
Example 3 pg 361
•
•
•
•
•
•
•
•
Solve (2x+1)(x-1)=14
We need to absorb the 14 into the stuff on the left
2x2-2x+1x-1= 14
2x2-x-15= 0
(2x+5)(x-3)=0
2x+5=0 or x-3=0
x=-5/2 or x=3
Plug them into the first equation and they both equal 14!
So the solutions are –5/2 and 3.
** Ex. 27-32**
Caution
• John? Why not just say (in the equation above:
(2x+1)(x-1)=14 that
2x+1=14 and x-1=14?
Because it doesn’t have the zero factor property to
help! Remember 6*1=6 as does 2*3=6 and
3*2=6 and 1*6=6. Which is which?
Might it not be 7*2=14? so 2x+1=7 or x-1=2?
Or the reverse? You see the problem.
Example 4 pg 360
•
•
•
•
•
Repeated Factor problem…
5x2-30x+45=0
5(x2-6x+9)=0
5(x-3)2=0 divide by 5
(x-3)2=0 you can take the square root of both
sides…
• x-3=0
• x=3 Even though x-3 is twice above, there is
only ONE solution here.
** Ex. 33-36**
Example 5 page 362
using grouping in your factoring
•
•
•
•
•
•
2x3-x2-8x+4=0
x2(2x-1)-4(2x-1)=0
(x2-4)(2x-1)=0
(x-2)(x+2)(2x-1)=0
x-2=0 or x+2=0 2x-1=0
x=2 or x=-2 or x= ½ So it has THREE solutions
(note the highest order term!)
** Ex. 37-44 **
Example 6a pg 363
1 2 1
x  x20
12
6
1 2 1

12 x  x  2   12(0)
6
 12

x 2  2 x  24  0
( x  6)( x  4)  0
x+6=0 or x-4=0
x=-6 or x=4
Example 6b
0.02x2-0.19x=0.1
100(0.02x2-0.19x)=100(0.1)
2x2-19x=10
2x2-19x-10=0
(2x+1)(x-10)=0
2x+1=0 or x-10=0
2x=-1
x=-1/2 or x=10
**Ex. 45-52**
Example 7 page 364
• After the decoding of the word problem we get the
expression x(2x+1)=55
• Expand it and lump on the left
• 2x2+x-55 =0
• (2x+11)(x-5)=0
• 2x+11=0 or x-5=0
• 2x=-11 or x=5
• x=-11/2 or x=5 We can’t have a negative width
so the solution we can use ifs 5 feet.
** Ex. 71-72 **
Enter the Pythagorean Theorem
• If you have a triangle then a2+b2=c2
Example 8 pg 365
Using the Pythagorean Theorem.
•
•
•
•
•
•
•
x2+(x+1)2=52
x2+ x2+2x+1=25
2x2+2x-24=0
x2+x-12=0
(x-3)(x+4)=0
x-3=0 or x+4=0
x=3 or x=-4 And since we are talking about
lengths, we can’t have a negative length, so the
working answer is 3.
**Ex. 73-74**
Example 5.6
•
•
•
•
•
Definitions Q1-6
Solve pre-factored equations Q7-10
Solve messier equations Q11-Q32
Solve slightly messier equations Q33-70
Word problems Q71-104
End for this week!