1. Polynomials Notebook.notebook October 02, 2012
Unit 2­ Polynomials POLYNOMIALS
At a birthday party, Crusty the Clown took the
age of the birthday girl and multiplied it by 6,
subtracted 2, multiplied it by 7, and then added
3. The resulting number was 493. How old is
the birthday girl?
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What is a polynomial??? A polynomial with 3 terms is
called a trinomial.
Monomial = polynomial with 1 term
Binomial = polynomial with 2 terms
A polynomial is made up of terms that are There is also a quadrinomial (4 terms)
& a quintinomial (5 terms) but we rarely
only added, subtracted or multiplied. use these words.
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The degree of a polynomial with only one variable is the
Find the degree of each polynomial.
largest exponent of that variable.
1. 3x2
The degree of this trinomial is 4. 2. 4x
y ‐1
2 3
3. 3x2 +4x2y3 ‐ 5z4
If the polynomial has more than one variable, you need to
look at each term. You find the degree of the polynomial
by adding the exponents of each term and naming the
largest power.
The degree of this trinomial is 3.
May 14­7:53 AM May 14­8:13 AM
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2. Polynomials Notebook.notebook October 02, 2012
The standard form of a polynomial is to put the terms
with the highest degree first.
Quiz Time!!!
Arrange the following in standard form: Just kidding...but here are a
1. 4x2 ‐ 2x + 3 ‐ 5x3 few questions...
2. ‐1 + x3 ‐ 9x2 + 8x
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May 14­8:32 AM May 14­8:34 AM
Adding &
Subtracting
Polynomials
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3. Polynomials Notebook.notebook October 02, 2012
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4. Polynomials Notebook.notebook October 02, 2012
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Remember, to add polynomials simply combine
like terms. To subtract polynomials, distribute
the -1 and then combine like terms.
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5. Polynomials Notebook.notebook October 02, 2012
10. 22.
Homework: 30. 46.
p. 113
#2- 58 eoe
May 14­11:30 AM May 14­11:31 AM
Answers to p. 113 #2­ 58 eoe
WARM UP
2. 4 6. 3
10. 1 14. ­10t + 8 Maya takes a number, multiplies it by
18. ­5a b ­ 5ab 22. ­5x ­ 6 4, subtracts 5, multiplies by 4 again
26. ­2x ­ 2x ­ 2 30. 3x ­ x ­ x ­ 6 and then adds 8. The resulting
34. 11x + 8 38. ­x + 4x + x ­ 5 number is 228. What number did
42. ­9x ­ 9x ­ 7 46. n ­ n + 1 Maya start with?
50. 3x + 7 54. ­3x ­ 2x + 10
58. 4x ­ 4
Sep 28­7:39 AM May 11­12:56 PM
To multiply 2 polynomials:
Multiplying Monomials • Multiply coefficients
• Add exponents of like bases
Examples:
Remem
ber th
1. 5(-7x) = you mul
tiply th
at whe
n
bases, e same
you AD
expone D the
nt. Don
2. -5y(-7x 2) = to mul
tiply co
't forg
et
efficien
ts!
3. 5x3y2(7xy4) =
4. 5xy(-9x2y3)(4x3y) =
5. 3x(-2x2y3)(-5x3y) =
May 11­12:56 PM May 14­11:31 AM
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6. Polynomials Notebook.notebook October 02, 2012
Can you put together the "old" stuff you learned
yesterday with the "new" stuff from today??? Power to a Power
• Distribute the outside exponent
Simplify:
to all inside exponents
OR
1. (x - 2y) - 4(-x + y) + 2(-4x + 5y)
• Rewrite what is inside the
parenthesis the same # as the
outside exponent and follow your
rules for multiplying polynomials.
2. 2m4n(7m5n3 + 6mn2) - 6mn2(2m4n - 7m5n3)
Jul 12­9:14 AM May 14­12:51 PM
Examples:
1. (3xy2)3 = Homework:
2. (-x4y5)4 =
p. 120 #6- 78
3. (-5a2b2c)3 =
Multiples of 6!
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Answers to p. 120 #6­ 78 (multiples of 6)
6. 12.
18. 24.
30. 36.
42. 48.
54. 60.
66. 72.
76.
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7. Polynomials Notebook.notebook October 02, 2012
Warm Up Multiplying Polynomials
Aidan spent $14.95 on ingredients for
cookies that will be sold at a bake sale.
If each cookie will sell for $0.40, how
many cookies must Aidan sell to earn a
profit?
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May 11­12:56 PM May 14­1:10 PM
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8. Polynomials Notebook.notebook October 02, 2012
Multiplying 2 Binomials
See next page for the process and answer...
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May 14­1:13 PM May 14­1:13 PM
A square has a side A rectangle has a length
represented by x -2. Find the represented by 5x + 3 and a
area of the square in terms of width represented by 2x - 7.
x. Find the area and write your
answer in standard form.
May 14­1:14 PM May 14­1:14 PM
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9. Polynomials Notebook.notebook October 02, 2012
A checker board is made up of
16 congruent squares. Each
square has a length
represented by x - 7.
Find the area of 1 square in
terms of x.
Then find the area of the
entire checker board in terms
of x.
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Homework:
p. 127
#2- 58 eoe
May 14­1:15 PM May 14­1:16 PM
Quiz Today
CR #2 due
tomorrow!
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10. Polynomials Notebook.notebook October 02, 2012
Warm Up Factoring
A hot air balloon begins ascending at the rate of 18
feet per minute. A second hot air balloon begins
ascending 8 minutes later at a rate of 27 feet per
minute. How many minutes will it take for the
second hot air balloon to catch the first hot air
balloon?
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What is a factor?
The numbers that are
multiplied together are called
factors of the final number.
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Greatest Common Factor
Try factoring these examples:
1. 28y2 - 4y =
2. 7x3 + 10x2 =
3. 12x3y4 - 39 x4y3 =
4. 5x + 7x2 + 9x4 =
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11. Polynomials Notebook.notebook October 02, 2012
Factoring the Difference
of 2 Perfect Squares
May 15­8:02 AM May 15­8:10 AM
Try factoring these examples:
1. x2 - 100 =
2. 9 - 25y2 =
3. 16x2 - 1 =
4. x6 - y10 =
5. 49x2 - 64y2 =
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Homework:
p. 135 #28- 40 even
p. 142 #1- 13 odd
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12. Polynomials Notebook.notebook October 02, 2012
Warm Up
The formula 180﴾n ­ 2﴿ is used to find the
sum of the interior angles of any
polygon. If n represents the number of
sides the polygon has, find the total sum
of the interior angles of an octagon.
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Factoring
Trinomials
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Here's a little trick Here's a little trick
(Bigger/ Both) (Bigger/ Both)
to help you figure out the signs. to help you figure out the signs.
What 2 #s multiply to What 2 #s multiply to
= (x )(x ) give you 12 but ADD to
= (x )(x ) give you 45 but
give you 7? SUBTRACT to
give you 4?
= (x 3)(x 4) = (x 9)(x 5)
Positive. Negative.
- for the letter i for
Therefore both = (x + 3)(x + 4) + for the letter t for Therefore the = (x - 9)(x + 5) the word "BIGGER."
signs are bigger # (9) gets
the word "BOTH." The bigger # gets the
positive. the - and the
Both signs are the same sign as the
same as the middle smaller # (5)
middle sign.
sign. gets the +.
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13. Polynomials Notebook.notebook October 02, 2012
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May 15­10:07 AM May 15­8:27 AM
Homework:
Worksheet
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14. Polynomials Notebook.notebook October 02, 2012
Warm Up
Akira scored 18 points in her last
basketball game, p of which came in the
4th quarter of the game. Write an
expression that can be used to find the
number of points she scored in the first 3
quarters.
May 15­8:18 AM May 11­12:57 PM
g by
Let's start out simple...Factor the following:
Factorin l pop
group ing wil to 1.
u go on
up if yo lculus!
tak e Ca
Factoring by 2.
Grouping
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KEY
PO
INT!
!!
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15. Polynomials Notebook.notebook October 02, 2012
The Arc Method
of Factoring
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Factor: The Arc Method of Factoring
5x2 ­ 8x + 3 Steps:
1. Take the coefficient in front of the x2 and multiply it by
Wait there are no 2 #'s that multiply to the last term.
give me 3 and add to give me 8!!! 2. Replace the last term with this new number.
(3)(1) = 3 3. Factor this new trinomial.
3+1=4 4. "Un­ arc" the factors by dividing the #'s by the original # in
from of the x2.
5. Reduce all fractions.
What am I supposed to do???? 6. If the fraction is not divisible, move the denominator in
from of your x term.
We will use the Arc Method of Factoring!
May 15­11:23 AM May 15­11:17 AM
= 5x2 - 8x + 3 5(3) = 15
= x2 - 8x + 15
= (x - 5)(x - 3) "Un- Arcing"
= (x - )(x - )
= (x - 1)(5x - 3)
Multiply to check this answer!!!
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16. Polynomials Notebook.notebook October 02, 2012
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Unit 2- Quiz #3
Homework:
today!
Worksheet
CR #2 due
tomorrow
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Warm Up Factoring Completely
The Drama Club is designing a set for an
upcoming show. The structure is shown in the
picture. Layer one has 1 triangle, but layer 2
consists of 3 triangles. How many triangles will
make up the 5th layer?
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17. Polynomials Notebook.notebook October 02, 2012
To factor x 3 + 7x2 + 12x you need to
Types of Factoring:
FACTOR COMPLETELY
1. GCF
Example: 3x2 + 9x3 + 12x = 3x ﴾1 + 3x + 4x﴿
4 2 2
Factoring completely is a three step
process:
2. Difference between 2 perfect squares 1. Factor a GCF from the expression, if
Example: y2 ­ 9 = ﴾y + 3﴿﴾y ­ 3﴿ possible.
2. Factor a Trinomial, if possible.
3. Trinomials 3. Factor a Difference Between Two
Example: x2 ­ 2x ­ 3 = ﴾x ­ 3﴿﴾x + 1﴿ Squares as many times as possible.
How would you factor x 3 + 7x + 12x???
2
x3 + 7x + 12x = x﴾x
2 2 + 7x + 12﴿
x﴾x + 3﴿﴾x + 4﴿
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Factor each of the following COMPLETELY: Factor each of the following COMPLETELY:
1. 2x ­ 8
2 1. 18x + 39x2 ­ 15x
3
2. n3 ­ 49n
2. 3x2 + 15x ­ 18
3. 25x2 ­ 100
May 15­1:55 PM May 15­1:55 PM
Homework:
Worksheet
Quiz tomorrow!!!
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18. Polynomials Notebook.notebook October 02, 2012
Warm Up Solving
Quadratic
What is the next
number in the Equations
sequence?
1, 2, 4, 8, 16, ...
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What is a quadratic?
A quadratic equation???
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It must get harder
than this, right??? What do we call the
highest power of a
polynomial????
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19. Polynomials Notebook.notebook October 02, 2012
Solve the equation below for x. How many values will
you end up with? Why?
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Here's an
interesting
fact...
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Remember the Arc Method!!!
3x2 + 13x - 10
x2 + 13x - 30
(x + 15)(x - 2)
(x + )(x - )
(x +5)(3x -2)
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20. Polynomials Notebook.notebook October 02, 2012
What do I do if
the quadratic
doesn't have 4
parts?
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May 16­8:33 AM May 16­8:33 AM
Here's a super
dooper hard one!
Homework:
Worksheet
Review tomorrow!
Test in 2 days =)
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21. Polynomials Notebook.notebook October 02, 2012
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