Transcript Polynomials

Math I, Sections 2.1 – 2.4
Standard: MM1A2c Add, subtract,
multiply and divide polynomials.
Today’s Question:
What are polynomials, and
how do we add, subtract
and multiply them?
Standard: MM1A2c.
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A monomial is a number, variable, or the
product of a number and one or more variables
with whole number exponents.
Whole numbers are 0, 1, 2, 3, …
The following are not monomials:
x
x-1
x1/2 2x
1
log x
ln x
y
x2
sin x
cos x
tan x
Monomial, Binomial, Trinomial - # of terms
Degree – add the exponents of each variable within
each term. The term with the highest sum defines
the degree of the expression.
Make a graphic organizer showing the possibilities
Degree
# Terms
Monomial
Binomial
Trinomial
Zero
First
Second
Third
Monomial, Binomial, Trinomial - # of terms
Degree – add the exponents of each variable within
each term. The term with the highest sum defines
the degree of the expression.
Make a graphic organizer showing the possibilities
Degree
# Terms
Monomial
Binomial
Zero
25
2+5
Trinomial 3 + 6 - 7
First
Second
x, 3y, 4z
3x2, 4xy
2x + 4,
2x2 – 6x,
3x + 5z
5xy + 3x
4x + y + 4z 3x2 – 4x + 6
Third
6x3, xy2
3x3 + 4x
2 + 9xy2
7x3 + 5X2 + 4
2xy2 + x2 + 4
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Start with 3x2 - 3 + 2x5 – 7x3
Put them in order, from largest degree to
smallest
Leading Coefficient
5
2x
–
3
7x
Degree
+
2
3x
-3
Discuss geometric representations of terms
1.
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“5” means 5 units, a rectangle 1 unit by 5 units
“2x” means a rectangle “x” units by 2 units
“x2” means a square “x” units by “x” units
“x3” means a cube with “x” on a side
Eliminate any parenthesis by distribution
property
Combine like terms
Make and play with algebra tiles:
2.
3.
4.
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Add (x2 + 2x – 3) to (x2 + 4x – 2)
Subtract (2x + 3) from (x2 + 3x + 5)
Subtract (2x + 3) from (x2 + x + 1)
Page 61, # 1 – 15 odds
(8 problems)
Get in groups of 2 to 3 and solve problem 16 on
page 61:
For 1995 through 2005, the revenue R (in dollars)
and the cost C (in dollars) or producing a
product can be modeled by
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1 2 21
R  t  t  400
4
4
1 2 13
C  t  t  200
12
4
Where t is the number of years since 1995. Write
an equation for the profit earned from 1995
through 2005. (hint: Profit = Revenue – Cost),
and calculate the profit in 1997.
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Go over problem 15 on page 61.
Domain – independent variable – you choose –
the x-axis.
Range – Dependent variable – you calculate –
the value depends on what you used for the
independent variable – the y-axis.
What is the domain and range of the problem?
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Multiply using distributive property.
NOTE: This always works!!!!!
Multiply (2x + 3) by (x + 2)
2x(x + 2) + 3(x + 2)
2x2 + 4x + 3x + 6
Answer: 2x2 + 7x + 6
Show how to multiply with algebra tiles
Show how to do it on the Cartesian Coordinate
Plane
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Multiply using distributive property.
NOTE: This always works!!!!!
Multiply (2x - 3) by (-x – 2)
2x(-x – 2) – 3(-x – 2)
-2x2 – 4x + 3x + 6
Answer: -2x2 – x + 6
Show how to multiply with algebra tiles
Show how to do it on the Cartesian Coordinate
Plane
Multiplication of more complicated expressions
are hard to show on algebra tiles and Cartesian
Coordinate plane.
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Example: Multiply x2 - 2x + 5 by x + 3 using the
distributive property
x(x2 - 2x + 5) + 3(x2 - 2x + 5 )
x3 - 2x2 + 5x +
3x2 - 6 + 15
x3 + x2 – x + 15
Show how to multiply on a table:
x2
x
+3
-2x
+5
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Example: Multiply x2 - 2x + 5 by x + 3 using the
distributive property
x(x2 - 2x + 5) + 3(x2 - 2x + 5 )
x3 - 2x2 + 5x +
3x2 - 6 + 15
x3 + x2 – x + 15
Show how to multiply on a table:
x
+3
x2
x3
+3x2
-2x
-2x2
-6x
+5
+5x
+15
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Example: Multiply x2 - 2x + 5 by x + 3 using the
distributive property
x(x2 - 2x + 5) + 3(x2 - 2x + 5 )
x3 - 2x2 + 5x +
3x2 - 6 + 15
x3 + x2 – x + 15
Show how to multiply on a table:
x
+3
x3
+x2
x2
x3
+3x2
-x
-2x
-2x2
-6x
+15
+5
+5x
+15
Page 66, # 3 – 18 by 3’s
and 19 – 22 all
(10 problems)
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You learned in Algebra 1A some patterns,
some special products of polynomials. You
need to remember/memorize them:
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
(x + a)(x + b) = x2 + (a + b)x + ab
(a + b)(a – b) = a2 – b2
NOTE: You can multiply these together by the
distribution property, but you will be required
to “go the other way” when we get to factoring
so please learn them now.
Page 70, # 3 – 18 by 3’s
(6 problems)
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So, what if we want to take a binomial and
multiply it by itself again and again?
Multiply (a + b)2
Multiply (a + b)3
Multiply (a + b)4
Multiply (a + b)5
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Shortcut:
Pascal’s Triangle
Multiply:
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(x + 2)5
(3 - x)4
(2x – 4)3
(3x – 2y)3
Page 75, # 12 – 17 all
(6 problems)