operations with monomials operations with polynomials
TRANSCRIPT
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Operations with MonomialsOperations with Polynomials
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Monomial• A number, a variable or a product of a number
and variable(s)
Examples: 1 -15 ½ x A y3
2x 5xyz -3x2 4xy3 ¼ x3
Not Monomials 2x + 4 4x – 5 8
x
6
4
x Isn’t this the same as
61( )
4x
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Quick ReviewQuick Review
34xbase
Exponent or power
3 = (4)(x)(x)(x)
the coefficent is not taken to t
What does it mean to raise to a power?
Example: 4x
he power
the x is multiplied times itself 3 times
Coefficient
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The Laws of Exponents:The Laws of Exponents:
Multiplicative Law of Exponents: When multiplying monomials and the bases are the same: multiply the coefficients and add the exponents .
m n m nx x x 3 4 3 4 7Example: x x x x
x x x x x x x 7x
Here is why this works…lets break the example down
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Example #1(3x2)(-6x3)
-18
Example #2(4y2z2)(y3)
1. Multiply the coefficients2. Write down the bases in alphabetical order
x
3. Add the exponents of the like bases
5
1. Multiply the coefficients
4
2. Write down the bases in alphabetical order
zy
3. Add the exponents of the like bases
5
4. Bring down any other exponents
4. Bring down any other exponents
2
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Practice
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The Laws of Exponents:The Laws of Exponents:
Exponential Law of Exponents: to raise a monomial to a power - raise the coefficient to the power - Multiply the exponents
323 x
627x
2 2 2(3 )(3 )(3 )x x x
3 2 2 2(3 )( )( )( )x x x
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The Laws of Exponents:The Laws of Exponents:
Exponential Law of Exponents: to raise a monomial to a power - raise the coefficient to the power - Multiply the exponents
42 32x y
16
1. 2 to the 4th power = 16
2. Write the variables down
3. Multiply each variables exponent time the exponent on the outside of the parentheses
yx8 12 yx
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Practice
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The Laws of Exponents:The Laws of Exponents:Division Law of Exponents: When dividing monomials: - simplify the coefficients - subtract the exponents with same bases - Simpify any Negative or Zero exponents
3
2
6 6
3 3
x x x x
x x x
2x
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The Laws of Exponents:The Laws of Exponents:Division Law of Exponents: When dividing monomials: - simplify the coefficients - subtract the exponents with same bases - Simpify any Negative or Zero exponents
2
4
2 2
3 3
y x x
y x x x x
2
2
3x
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2
5
8x y
4x y
2
3x 0y3
2
x3 2
3 7
12x y
3x y
4
0x 5y
5
4
y5
8
4xz
16yz
1
4
x 3z
3
x
4z
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Practice
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Binomial – 2 monomials connected by addition or subtraction
Trinomial – 3 monomials connected by addition or subtraction
Polynomial – 1 or more monomials connected by addition or subtraction
2 2x 24 5x y
22 2 5x x
2 2x 22 2 5x x 4 37 3 5 9x x x
Like Terms – have the same variable and the same exponent
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Add Polynomials – ignore the parentheses and combine like terms
1.D2 2 2(3 2 5 ) (4 4 3 )x x x x x y 2 2 23 2 5 4 4 3x x x x x y
1. Combine the term with the biggest exponent2. Then combine the terms with next biggest exponent3. Continue combing terms in descending order
2x x 3y
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Subtract Polynomials – distribute the negative 1 and combine like terms
7.D 2 2 2(3 2 5 ) (4 4 3 )x x x x x y 24 4 3x x y
1. Distribute the negative 1 (turn everything to its opposite)2. Cross out the 2nd parentheses of original problem3. Combine like terms, going in descending order
29x 9x 3y
1
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Add Polynomials – ignore the parentheses and combine like terms
1. Combine the term with the biggest exponent2. Then combine the terms with next biggest exponent3. Continue combining terms in descending order
Subtract Polynomials – distribute the negative 1 and combine like terms
1. Distribute the negative 1 (turn everything to its opposite)2. Cross out the 2nd parentheses of original problem3. Combine like terms, going in descending order