Download CHAPTER 7: POLYNOMIALS Quiz results Sum Difference Product

10/4/2010
Quiz results
ƒ Average 73%: high score 100%
ƒ Problems:
Sections 7.1, 7.2:
Sums, differences, products of polynomials
CHAPTER 7: POLYNOMIALS
à Keeping track of negative signs
‚ —x—=+
‚—÷—=+
à Function notation
‚ f(x) ~ y: the result of the input
‚ x is the input
ƒ Retakes can be done up to 11 AM Oct. 11
Sum
Difference
ƒ The result of adding
ƒ Subtraction
ƒ The sum of two positive values is a
ƒ Needs to be done in order from left to
positive
ƒ The sum of two negative values is a
negative
ƒ When the values are different signs
à Find the difference
right
ƒ Subtract a negative value is the same as
adding a positive
ƒ Suggest you change signs of terms that
are subtracted and use adding rules
à Use the sign of the one with the larger
absolute value
Product
Term
ƒ The result of multiplication of factors
ƒ A constant, a variable or a product of a
ƒ Product of two positives is positive
constant and one or more variable
factors
ƒ Terms
T
are separated
t d ffrom one another
th b
by
addition
ƒ Subtraction in an expression or equation
needs to be interpreted as adding a
negative term:
ƒ Change its sign and use adding rules.
ƒ Product of two negatives is positive
ƒ If signs are opposite, the product is
negative
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Monomial
Polynomial
ƒ Single term
ƒ Monomial or sum of monomial terms
ƒ May be product of a constant and one or
ƒ Named by number of terms
more variables
ƒ Variables may be raised to powers
(exponents)
ƒ Absence of a numeric constant implies
the constant is 1
à Binomial: 2
à Trinomial: 2
Polynomial degree
Terms are made of factors
ƒ The greatest exponents in any term
ƒ Numeric factor: Coefficient
ƒ Add up the values of the exponents in
ƒ Variable factors represented by letters
each term
à 4x3y2+2x2y2+9xy3
à 1st term: 5th degree
à 2nd term: 4th degree
à 3rd term: 4th degree
ƒ Fifth degree polynomial
“Like Terms”
Quadratic expression or function
ƒ Have exactly the same variable set
ƒ Second degree polynomial
ƒ Combine the coefficients (numeric
ƒ f(x) = 2x2 + 4x – 3
factors) by addition
ƒ Recall that the term has a sign
ƒ Recall f(x) is the function, equivalent to
the output
à Positive
ƒ x is the input: example x=3
à Negative
ƒ f(3) = 2(3)2 + 4(3) – 3 = 2(9) +12 – 3 = 27
à Combine by addition rules
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f(x)
30
Quadratic function
Cubic function
25
80
60
20
ƒ Graph is called a parabola
15
ƒ Vertex:
10
à this has minimum value
shape
0
-6
-4
à Have maximum value
-2
40
20
ƒ May have horizontal
5
ƒ Axis of symmetry
ƒ Some open downward
ƒ Graph is a serpentine
0
2
4
line that
li
th t crosses
graph in more than
one place
x
-5
0
6
-6
-4
4
-2
2
0
2
4
6
-20
-40
-10
-60
-80
Adding functions
Subtracting functions
ƒ f(x) =
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
2x2 + 4x — 3
g(x) = —3x2 + 9x — 7
(f+g)(x) = 2x2 + 4x — 3 —3x2 + 9x — 7
Add like terms: watch the signs!!
Do it vertical, not on one line
2x2 + 4x — 3
+ —x2 + 9x — 7
—x2 +13x — 10
Multiplying monomials
(finding products)
ƒ Multiply numeric coefficients
ƒ Combine exponents on like variables
à x2 · x3 =
à Write factors without exponents to ‘see’ what
the exponent means
à x·x · x·x·x = x5
ƒ
ƒ
ƒ
ƒ
f(x) = 2x2 + 4x — 3
g(x) = —3x2 + 9x — 7
(f—g)(x) = 2x2 + 4x — 3 — (—3x2 + 9x — 7)
Ch
Change
every sign
i off ffunction
ti b
being
i
subtracted!! Then add, watching signs
Do it vertical, not on one line
2x2 + 4x — 3
+ +3x2 — 9x + 7
5x2 —5x + 4
Multiplying polynomial by
monomial
ƒ 2(3+5) = 2(8) = 16, right?
ƒ “Distribute” multiplication over addition
ƒ 2(3) +2(5) = 6 + 10 = 16, same thing
ƒ 2x(3x2 + 5x) = 6x3 + 10x2
à same thing with variables,
à just now terms are not ‘like’ so you cannot
combine them
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Product of binomials
Product of binomials
ƒ (2+3)(2+5)=(5)(7)=35
ƒ F.O.I.L. method
ƒ Label terms
à Firsts, Lasts, Outsides, Insides
à F L F L
ƒ Multiply second factor by each term of
first factor, then distribute
ƒ (2+3)(2+5)=2(2+5)+3(2+5)=14+21=35
ƒ With variables: (x+3)(x+5)=
ƒ x(x+5)+3(x+5)=
ƒ x2+5x+3x+15=
ƒ Combine like terms: x2+8x+15
(x+3)(x+5)
ƒ
à O I
I
O
ƒ Multiply: draw line, write product, draw, write…
à
à
à
à
Firsts x2
Outsides +5x
Insides +3x
Lasts +15
ƒ Combine like terms: x2+8x+15
Products of higher degree
polynomials
ƒ Be very methodical
ƒ Draw line for the product of two terms
ƒ Write the product of those terms
ƒ Draw another line for product of terms
ƒ Write the product of those two terms
ƒ Etc: DO NOT DRAW ALL THE LINES AND GO
BACK TO FIND THE PRODUCTS!!
ƒ YOU WILL GET LOST!!
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