Ch 10 Alg 1 07-08 ML, AS

Report
Chapter 10:
Polynomials
and
Factoring!!!
By Anastasia
Stocker
& Matthew Laredo
Key Words
• Monomial-
a number and one or more variables with
whole number exponents.
• Degree of a monomial in one variable-
the
sum of the exponents of the variables in the monomial.
• Polynomial-is a monomial or a sum of monomials (x’s with
different degrees)
• Binomial-a polynomial of two terms
• Trinomial-polynomial of three terms
• Standard form-the terms are arranged in decreasing
order, from largest exponent to smallest exponent.
• Degree of polynomial in one variable-the
largest exponent of a variable.
Monomials
• Remember a monomial
is just a number. It
could or could not
have an X or Y in it.
• The “degree of the
monomial” is simply
the sum of the
exponents. A.K.A. add
the little numbers
above the X’s and Y’s
• Examples of
monomials:
8, -2x, 3x²y or ½x²
• Therefore the
“degree of the
monomials” would be 0,
1, 3 and
2…respectively.
• The degree of 8 is
zero because 8 = 8xº.
• NOTE: Any term to
the 0 power equals 1
i.e. xº=1
Section 10.1
POLYNOMIALS!!!
Adding
(5x³ + x² -2x + 7) + (3x² – 4x + 7)
AND
Subtracting
(3x² – 5x + 3) – (2x² – x - 4)
5x³ + x² - 2x + 7
+
3x² – 4x + 7
5x³ + 4x² - 6x + 14
3x² – 5x + 3
–
2x² + x + 4
x² – 4x + 7
Adding & Subtracting
Polynomials
Remember when dealing with
“polynomials”, aka numbers, group like
terms. Put the x’s with the x’s and
the x²’s and with x²’s and so on and
so forth. ALSO!!!! Do not forget to
distribute the subtraction sign within
the parentheses. Oh, and these are
parentheses ( ).
Section 10.2
Multiplying Polynomials
Using FOIL:
Product of First terms + Product of
Outer terms+ Product of Inner terms+
Product of Last terms
(3x+4)(x+5)=3x²+15x+4x+20
Then combine like terms!!
=3x²+19x+20.
Using the Punnett Square
• Take the terms
inside of the
parenthesis and
put them on either
side of the
squares.
• Then multiply to
fill the boxes
inside of the
square.
• For the final
answer combine
like terms (same
colors)
3x + 4
X 3x² 4x
+ 6x 8
2
=3x² + 10x + 8
Multiply Binomials to Find
an Area
The window has a total
height of 3x+10 and
total width of 2x+6.
The are of the window
is represented by the
product of the height
and width.
5
3x
• A= height ·width
• A= (3x+10)(2x+6)
• =6x²+18x+20x+60
• =6x²+38X+60
5
3
2x
3
Section 10.3
Special Products
of Polynomials
SUM and DIFFERENCE PATTERN
(a+b)(a-b)= a² - b²
for example: (3x-4)(3x+4)= 9x² - 16
SQUARE OF A BINOMIAL PATTERN
EXAMPLE:
(a+b)² = a² + 2ab + b²
(a – b)(a +b)= a² + b²
(a+b) ²= a² - 2ab + b²
(5t-2)(5t + 2)= (5t) ² - 2²
= 25t² - 4
To apply: 1st Write Pattern
2nd Apply pattern
3rd Simplify.
USE THE SQUARE OF A BINOMIAL PATTERN
Find the product.
(3n + 4)²
Write equation.
(a + b) ² = a² + 2ab + b²
Write pattern.
(3n + 4)²= 3n² + 2(3n)(4) + 4²
Apply the pattern using
parenthesis.
= 9n² + 24n + 16
Simplify (mulitply out).
Finding the Area of Figure
To find the area of a particular region within a figure 1st
set the area of the region you want to FIND equal to 2nd
the entire are of the figure. 3rd subtract the area of the
region you do not want in total area. 
Section 10.4
Solving Quadratic Equations in Factored
Form
The Zero Product Property:
Solving the equation (x-2)(x+3)=0
Set each factor to zero so that
Solve for X
x–2=0 &
X=2
x+3=0
X = -3
A polynomial is in factored form
when “it is written as a product of
two or more factors”….
Basically when an equation
has parentheses ( ),
separating “terms”...things
with the X’s or Y’s
x² + 5x + 6 = 0
Not Factored

(X + 2) (X + 3) = 0
FACTORED!!!!!
☻
A “repeated-factor” equation is when the equations
factors are the same. Meaning the factors of (x+3)²
are (x+3)(x+3). Therefore…when solving these
equation, use the zero product property.
(x + 5) ² = 0
X+5=0
X = -5
When solving a
factored cubic
equation, again…use
the same method.
i.e. (x+3)(x+4)(x+5)= 0
Graphing a factored equation???
NO PROBLEM!!!
FIRST: Find the x-intercepts by
solving the equation for X….like we
just showed you how to.
Second: Take the average of
the x-intercepts to find the xcoordinate of the vertex
Third: Plug the xcoordinate back into the
original equation to find the
vertex
Fourth: Use the vertex and
x-intercepts to sketch the
graph
y = (x + 2)(x - 3)
X = -2 & 3
3 +(-2)
2
=
½
y = (½ + 2) (½ - 3)
y = - 25
4
Figure it out, I cant
hold your hand
through everything
Section 10.5
FACTORING
x² + bx + c
To factor a trinomial of this form
means to write the product of the
two binomials (factored form)
x²+9x+14=(x+2)(x+7)
TRINOMIAL
BINOMIAL
(FACTORED
FORM)
Remember: b & c are integers (numbers)
How to Factor:
When b and c are positive
Factor x²+6x+8.
What p and q could be:
1, 8
b=6 and c=8, so,
therefore you need
to find numbers
where p times q is 8
and when added to
together they equal
6.
2, 4
What p+q would equal:
1+8= 9, which does not = 6
2+4=6, since 6=6 you need
these numbers!
The Answer:
x²+6x+8 = (x +2)(x +4)
What about when there are
negatives???
When negative numbers are added the same steps
need to be taken to factor your equation.
Just remember:
p and q must multiply to equal c and add
up to equal b
Section 10.6
Factoring ax² + bx + c
Factors of 6
Example: 6x² + 22x + 20= (3x+5)(2x+4)
b=12+10= 22
Factors of 20
To factor ax² + bx + c is to find m and n whose product is a and
numbers p and q whose product is c so that the middle term is
the sum of the OUTER and INNER products of FOIL.
mxn=a
ax² + bx + c = (mx+p)(nx+q)
pxq=c
b= mq+np
Example: factoring
ax² + bx + c
Factor 3x² + 5x – 2
For this trinomial, a=3, b=5, c= -2.
Because c is negative, you need
to find numbers p and q with
different signs.
•
Write the numbers m and n
whose product is 3 and the
numbers p and q whose product
is –x.
•
Use these numbers to write
trial factors. Then use the
Outer and Inner products of
FOIL to check the middle
term.
m and n
1, 3
Trial factors
(x-1)(3x+2)
(x+1)(3x-2)
(x+2)(3x-1)
Answer- 3x² + 5x – 2= (x + 2)(3x – 1)
p and q
-1,2
1,-2
Middle term
2x-3x=-x
-2x+3x=x
-x+6x=5x
Section 10.7
Factoring Special Products
Difference of Two Squares Patterns
EXAMPLE:
a² - b² = (a - b) (a - b)
9x² - 25 = (3x + 5) (3x - 5)
Perfect Square Trinomial Pattern
a² + 2ab + b² = (a + b) ²
EXAMPLE: x² + 14x + 49 = (x + 7) ²
a² - 2ab + b² = (a - b) ²
EXAMPLE: x² - 12x + 36 = (x - 6) ²
Section 10.8
Factoring Cubic Polynomials
When factoring a cubic
polynomial, you should factor out
the greatest common factor
(GCF) first and then look for
other patterns.
Example: factor the
greatest common factor
out of 14x³-21x².
1st find the GCF of 14x³
and 21x².
14x³= 2· 7 · x · x · x
21x²= 3 ·7 ·x ·x
GCF= 7 ·x ·x= 7x²
Then use the distributive property to factor out the greatest common
factor from each term.
Answer- 14x³-21x²= 7x²(2x-3).
Other ways to Factor
Factor Completely
Factor 4x³ + 20x² + 24x
Solution:
4x³ + 20x² + 24x=
4x(x²+5x+6) Factor out GCF
= 4x(x+2)(x+3) Factor trinomial
Factor by Grouping
Factor x³-2x²-9x+18
completely.
Solution:
x³-2x²-9x+18=(x³-2x²)+
(-9x+18)
=x²(x-2)+(-9)(x-2)Factor each group
=(x-2)(x²-9)
Use distributive
=(x-2)(x-3)(x+3)
property.
Factor
difference of two
squares.
AND NOW YOU KNOW!!!
THE END
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