j. baker february 20101. 2 monomials: a number, variable, or the product of quotient of a number and...
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J. Baker February 2010 1
J. Baker February 2010 2
• 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 or 2 B x2
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ExamplesMonomials NOT monomials
12 A + b
X 5 – 7d + 2f
-4x2
11ab1/3 xyz2
5
w2
Classify the following in a table on the next page:
4x – 7-9X2 +2xy +y2
14m2abc2 + 13x 5
3x2 -11xyA + 2b + 4c2x + 9y3y2 3a – 7b2 -4c2x2 + 5x + 4
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Classify by number of terms Monomial Binomial Trinomial
Degree of a monomial: The sum of the exponents of its variables
8y3 3
4y2 a1b1 4
-14 0
42a1b1c1 3
+
Hint: variableBy themselvesHave a degreeOf 1
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Degree of a polynomial with more than one term:The GREATEST degree of all terms
3x2 + 8a2b + 4 2, 3, 0 3
7x4 + 4x - 9x2y7 4, 1, 9 9
Polynomial degree of terms degree of polynomial
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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)
Descending: exponents of specified variable go from largest to smallest (down)
Example: 1.
2.
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
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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
(x2 + y2) – (-x2 + y2)= x2 + y2 + x2 – y2
= 2x2
(2a2 – ab + b2) – (3a2 + 5ab – 7ab2)
= 2a2 – ab + b2 – 3a2 – 5ab + 7ab2
= -a2 -6ab + b2 + 7ab2
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Finding Perimeter: add all sides(add coefficients and leave variables the same)
Square
-3x2
-3x2 + -3x2 + -3x2 + -3x2
= -12x2
2a
-6b2
-3a + b2
2a + -6b2 + -3a + b2
= -a -5b2
-m + 5n
2m – 4n
2m - 4n + -m + 5n + 2m - 4n + -m + 5n
= 2m + 2n
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Laws of Exponents
Review: monomial = 1 term (no + or - )
2x5
coefficient variable
exponent
Product of Power: “multiplying with exponents”
General rule: am * an = am+n 1. b3 * b8 = b11
2. x (x3) (x4) = x8
3. (a2b) (a5b4) = a7b5
4. (6c2d3) (5c4d) = 30c6d4
5. (10w2) (-7w2y3) = -70w4y3
6. (a2b) (3a5b6) = 3a7b7
7. (-ab2) (5a2) (-b3) = -5a3-b5
Add the exponents
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Power of a Power
General rule: (am)n = amn [multiply the exponents]
1. (x5)2 = x10
2. (a2)6 = a12
3. (b)8 = b8
4. (c5)4 = c20
Power of a Product
General Rule: (ab)m = ambm
(backwards distributing)
1. (ab)4 = a4b4
2. (c2d3)5 = c10d15
3. (-2x3)4 = 16x12
4. (-9ax3y2)3 = -729a3x9y6
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Mix it Up! Use all your rules!
1. (2d)2(5d3)
2. -2x(6x2)3
3. (-3ab)3(2b3)
4. (3x)2(4x2yz6)
5. (3a)(-a2b)2(4ab2c)
6. -5(2p2)3
7. (-3ab)(-3ab3)(-3a2b4)3
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Quotient of Powers: dividing with exponents
General rule: am = am-n
an
Subtract the exponents
X5
X2
4x9
x3
m4w3
mw2
10m4
30m
50x5y3
-25x2y= X5-2 = X3
= 4x9-3 = 4x6
= m4-1 w3-2 = m3w
= 10 m4-1 = 1 m3
30 3
= 50 x5-2 y3-1
-25= -2x3y2
Reminder: variables that stand alone
have an exponent of 1
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Zero ExponentB4
B4 = B4-4 = B0 = 1 Think of it as 5 or -10 or B4
5 -10 B4
General Rule: a0 = 1
ANYTHING to the zero power = 1
Negative ExponentN3
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 1 = bn
an b-n
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Mix it up!
1. 3x-2
2. -5a2b-3
3. 5a-1
10b-2
4. 15a-3
45a-2
5. (6a-1b)2
(b2)4
6. 24w3t4
6w7t2
7. 15x3
5x0
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Distribute monomial into polynomial– Multiply coefficients– Add exponents
Multiplying Monomial x Polynomial
3(2x - 5) = 3*2x – 3*5= 6x -15
7b( 4b2 – 18)
7b( 4b2 – 18)= (7*4 b1+2) – (7b * 18)= 28b3 – 126b
3d(4d2-8d-15)
3d(4d2-8d-15)= (3*4 d1+2) – (3*8 d1+1) – (3*15 d1)= 12d3 – 24d2 – 45d
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Binomial * Binomial FOIL
F first: multiply the first monomials in each set of ()
A*CO outside: multiply monomials
on the outside of each set ().
A*DI inside: multiply monomials on
the inside of each set ().
B*CL last: multiply monomials in the
last position in each set ().
B*D
(2x * 4x) + (2x * 4) + (3 * 4x) + (3 * 4)
8x2 + 8x + 12x + 12
8x2 + 20x + 12
(A + B)(C + D)
(A*C) + (A*D) + (B*C) +(B*D)
(2x +3)(4x +4)
Combine like terms if possible
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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)
8x2
1
12x2
8x3
124
(2x + 3) (4x + 4)
2x +3
4x
+4
Combine like terms (boxes 2 and 3)
8x2 + 8x + 12x + 12
8x2 + 20x + 12
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Polynomials: rules to remember
COEFFICIENTS(number in front of variable)
EXPONENTS
ADDing Add coefficients of like terms
Leave exponents and variables alone. They are
used only for combining like terms in this step
SUBTRACTing-distribute the negative
-add
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(outside parentheses)
Raise coefficient to the power shown outside
parenthesis
Multiply exponent inside parenthesis with the exponent outside the
parenthesis