Transcript All about polynomials booklet

J. Baker February 2010
1
• Monomials: a number, variable, or the
product of quotient of a number and
variable.
• Polynomial: a monomial or the sum of 2 or
more monomials
– Types of Polynomials
• Monomials (1 term)
• Binomials (2 terms)
• Trinomials (3 terms)
BEWARE: a quotient with a variable in the denominator is NOT a polynomial!
Example:
A
B
or
2
x2
J. Baker February 2010
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Examples
Monomials
NOT monomials
12
X
-4x2
11ab
1/ xyz2
3
A+b
5 – 7d + 2f
5
w2
Classify the following in a table on the next page:
4x – 7
-9
X2 +2xy +y2
14m
2abc
2 + 13x
5
3x2 -11xy
A + 2b + 4c
2x + 9y
3y2
3a – 7b2 -4c
J. Baker February 2010
2x2 + 5x + 4
3
Classify by number of terms
Monomial
Binomial
Degree of a monomial:
Hint: variable
By themselves
Have a degree
Of 1
Trinomial
+
The sum of the exponents of its variables
8y3
3
4y2 a1b1
4
-14
0
42a1b1c1 J. Baker February 2010 3
4
Degree of a polynomial with more than one term:
The GREATEST degree of all terms
Polynomial
degree of terms
degree of polynomial
3x2 + 8a2b + 4
2, 3, 0
3
7x4 + 4x - 9x2y7
4, 1, 9
9
J. Baker February 2010
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Ordering Polynomials: the terms are written
in ascending or descending order with respect to one
variable’s exponents
Ascending: exponents of the specified
variable go from smallest to largest (up)
Example:
1.
2.
Descending: exponents of specified variable
go from largest to smallest (down)
Example:
1.
2.
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Adding polynomials: add the coefficients
of like terms and keep the variables and exponents
the same Parenthesis are not necessary in this
step, the simply separate terms to be added
1, (4x + 6y) + (3x + 9y) = 7x + 15y
2. (3x2 – 5xy + 8y2) + (2x2 + xy – 6y2)
= 5x2 – 4xy + 2y2
3. (3p2 – 2p + 3) = (p2 + 7) = 4p2 – 2p + 10
J. Baker February 2010
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Subtracting Polynomials:
* Distribute the negative to everything behind it
* change the “ – “ to “ + “ and add as before
(7x2 – 8) – (-3x2 +1)
= 7x2 – 8 + 3x2 -1
= 10x2 – 9
(2a2 – ab + b2) – (3a2 + 5ab – 7ab2)
= 2a2 – ab + b2 – 3a2 – 5ab + 7ab2
(x2 + y2) – (-x2 + y2)
= -a2 -6ab + b2 + 7ab2
= x2 + y 2 + x2 – y2
= 2x2
J. Baker February 2010
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Finding Perimeter: add all sides
(add coefficients and leave variables the same)
Square
-3x2 + -3x2 + -3x2 + -3x2
= -12x2
-3x2
-m + 5n
2a + -6b2 + -3a + b2
-6b2
2m – 4n
= -a -5b2
2a
2m - 4n + -m + 5n + 2m - 4n + -m + 5n
-3a +
b2
= 2m + 2n
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Review: monomial = 1 term (no + or - )
5
2x
coefficient
exponent
variable
Laws of Exponents
Product of Power: “multiplying with exponents”
General rule: am * an = am+n
1.
2.
3.
4.
5.
6.
7.
b3 * b8 =
x (x3) (x4) =
(a2b) (a5b4) =
(6c2d3) (5c4d) =
(10w2) (-7w2y3) =
(a2b) (3a5b6) =
(-ab2) (5a2) (-b3) =
b11
x8
a7b5
30c6d4
-70w4y3
3a7b7
3-b5
J. Baker February-5a
2010
Add the exponents
10
Power of a Power
General rule: (am)n = amn [multiply the exponents]
1.
2.
3.
4.
(x5)2 =
(a2)6 =
(b)8 =
(c5)4 =
x10
a12
b8
c20
Power of a Product
General Rule: (ab)m = ambm
(backwards distributing)
1. (ab)4 =
2. (c2d3)5 =
3. (-2x3)4 =
4. (-9ax3y2)3 =
a4b4
c10d15
12
16x
J. Baker February 2010
-729a3x9y6
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Mix it Up! Use all your rules!
1.
2.
3.
4.
5.
6.
7.
(2d)2(5d3)
-2x(6x2)3
(-3ab)3(2b3)
(3x)2(4x2yz6)
(3a)(-a2b)2(4ab2c)
-5(2p2)3
(-3ab)(-3ab3)(-3a2b4)3
J. Baker February 2010
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Quotient of Powers: dividing with exponents
General rule: am = am-n
an
X5
5-2 = X3
=
X
X2
4x9
x3
50x5y3
-25x2y
= 50 x5-2 y3-1
-25
= -2x3y2
= 4x9-3 = 4x6
m4w3
4-1 w3-2 = m3w
=
m
mw2
10m4
30m
Subtract the exponents
= 10 m4-1 = 1 m3
30
3 J. Baker February 2010
Reminder:
variables that stand alone
have an exponent of 1
13
Zero Exponent
B4 = B4-4 = B0 = 1
B4
Think of it as 5 or -10
5
-10
or B4
B4
General Rule: a0 = 1
ANYTHING to the zero power = 1
Negative Exponent
N3
N7
= n-4 ***cannot have negative exponents***
To remove a negative exponent move only the number or variable up or down
General Rule: a-n = 1
or
aJ.nBaker February 2010
1 = bn
b-n
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Mix it up!
1.
3x-2
5. (6a-1b)2
(b2)4
2. -5a2b-3
3. 5a-1
10b-2
6. 24w3t4
6w7t2
7. 15x3
5x0
4. 15a-3
45a-2
J. Baker February 2010
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Multiplying Monomial x Polynomial
Distribute monomial into polynomial
– Multiply coefficients
– Add exponents
3(2x - 5) = 3*2x – 3*5
= 6x -15
7b( 4b2 – 18)
3d(4d2-8d-15)
7b( 4b2 – 18)
= (7*4 b1+2) – (7b * 18)
= 28b3 – 126b
3d(4d2-8d-15)
= (3*4 d1+2) – (3*8 d1+1) – (3*15 d1)
= 12d3 – 24d2 – 45d
J. Baker February 2010
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Binomial * Binomial FOIL
F first: multiply the first
monomials in each set of ()
(A + B)(C + D)
A*C
O outside: multiply monomials
(A*C) + (A*D) + (B*C) +(B*D)
on the outside of each set ().
A*D
I inside: multiply monomials on
the inside of each set ().
(2x +3)(4x +4)
B*C
L last: multiply monomials in the
(2x * 4x) + (2x * 4) + (3 * 4x) + (3 * 4)
last position in each set ().
B*D
Combine like terms if possible
J. Baker February 2010
8x2 + 8x + 12x + 12
8x2 + 20x + 12
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Binomial * Binomial Box
Place binomials on either side of
the box and multiply each box
(Similar to Punnett squares in
biology)
4x
(2x + 3) (4x + 4)
2x
+3
8x2
12x
1
2
8x
12
3
4
Combine like terms (boxes 2 and 3)
+4
8x2 + 8x + 12x + 12
8x2 + 20x + 12
J. Baker February 2010
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Polynomials:
COEFFICIENTS
rules to remember
(number in front of variable)
ADDing
Add coefficients of like
terms
Leave exponents and
variables alone. They are
used only for combining like
terms in this step
SUBTRACTing
First set of ( ) never change
Distribute the negative to
everything behind it
Change “-” into “+” and add
as normal
Leave exponents and
variables alone. They are
used only for combining like
terms in this step
MULTIPLYing
Multiply numbers as normal
Add exponents of like terms
DIVIDing
Divide numbers as normal
Subtract exponents of like
terms. No Negative
exponents allowed
Exponents
Raise coefficient to the
power shown outside
parenthesis
J. Baker
February 2010
-distribute the negative
-add
(outside parentheses)
EXPONENTS
Multiply exponent inside
parenthesis with the
exponent outside the 19
parenthesis