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Algebraic Operations
Algebraic OperationsGoals• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
• Identify monomial, binomial, and polynomial terms in an algebraic expression.
Big Idea
Algebraic expressions are often a sum of terms. An expression can be simplified by combining like terms.
Algebraic OperationsGoal• Identify monomial, binomial, and polynomial terms in an algebraic expression.
• Describe the order of the terms in a polynomial as ascending or descending.
Big Idea
A polynomial is an algebraic expression involving a sum of terms. Polynomial with 1, 2 or 3 terms are given special names:
• monomial - 1 term e.g. 5x or 5 or 3abc
• binomial - 2 terms e.g. 1 + x or 3x2 + 2
• trinomial - 3 terms e.g. 1 + x + x2 or ab + bc + 3
Algebraic OperationsGoal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
Big Idea
A polynomial can be simplified by combining like terms. Like terms have variables with the same power.
Remember x0 = 1
Algebraic Operations
HW Due Tuesday 11/23176.2, and any 4 of 176.3-26, 181.5, any 3 of 181.19-29, and any 4 of 181.30-47, 181.48, 181.51, any 5 of 181.52-73
Algebraic OperationsGoalExpress the product of terms with the same base using a sum of exponents.
Big Idea
xa • xb = xa+b
22 • 2 = 23
ab2 • a = a2b2
For example:
Algebraic Operations
xa • xb = xa+b
32 • 3 =x2 • x =x2 y • x y =x2 y • x y2 =
Apply the rule:
Algebraic OperationsGoal• Express the power of a power using a product of exponents.
Big Idea(xa )b= xab
22 • 2 = 23
ab2 • a3 = a4b2
Compare with:
(22)3 = 26
(ab2)3 = a3b6
For example:
Algebraic Operations(xa )b= xab
(32)3 =(xy2)2 =x • (xy)2 =xy2 • (xy)3 =
Apply the rule:
HW Due Tuesday 11/23176.2, and any 4 of 176.3-26
HW Due Wednesday 11/24181.5, any 3 of 181.19-29, and any 4 of 181.30-47, 181.48, 181.51, any 5 of 181.52-73
Algebraic OperationsGoal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
Big IdeaMonomials can be multiplied:• Group variables and numerical factors• Multiply the numerical factors• Multiply the variables
For example:
6x2•(-3x4)=6•(-3)•x2•x4 =-18 x6
Algebraic OperationsBig IdeaMonomials can be multiplied:• Group variables and numerical factors• Multiply the numerical factors• Multiply the variables
3x2•2x4=
x2y•2x=
x3y• xy=
5
2
5
2
3
5
Algebraic OperationsGoal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
Big IdeaBinomials can be multiplied by monomials:• Group variables and numerical factors• Distribute the monomial over the sum• Multiply the numerical factors• Multiply the variables
For example:
6x2•(-3x4+2)=6•(-3)•x2•x4 + 6x2•(2)=-18 x6 + 12x2
Algebraic OperationsBig IdeaBinomials can be multiplied by monomials:• Group variables and numerical factors• Distribute the monomial over the sum• Multiply the numerical factors• Multiply the variables
x2• (2+x)=
ab • (3a + b) =
x2y • (3xy + 2) =
5
2
Algebraic OperationsGoal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
HW Due Wednesday 11/24181.5, any 3 of 181.19-29, and any 4 of 181.30-47, 181.48, 181.51, any 5 of 181.52-73
Algebraic Operations
HW Due Monday 11/29185.2, 185.21, 185.22, 185.24, 185.25, 185.28, 185.29, 185.32, 185.35, 185.36 and 185.43
Goal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
Algebraic Operations
Big IdeaPolynomials can be multiplied by monomials:• Group variables and numerical factors• Distribute the monomial over the sum• Multiply the numerical factors• Multiply the variables
For example:
(6x2 ) •(-3x4+2)=6•(-3)•x2•x4 + 6x2•(2)=-18 x6 + 12x2
Try these:
(−2)(+6cd)(−e) =
(18r5)(−5r2 ) =
(+6x2y3)(−4x4y2 ) =
Try these:
−16(34
c−58
d) =
5r2s2(−2r2 + 3rs−4s2 ) =
−8(2x2 −3x−5) =
Try these:
4(2x +5)−3(2−7x) =5x(2 −3x) −x(3x−1) =
3a−2a(5a−a) + a2 =7x+ 3(2x−1)−8 =y(y+ 4)−y(y−3)−9y=
Algebraic OperationsBig IdeaPolynomials can be multiplied by polynomials:• Group variables and numerical factors• Distribute the monomial over the sum• Multiply the numerical factors• Multiply the variables
For example:
(6x2 -1) •(-3x4+2)=6•(-3)•x2•x4 + 6x2•(2) + (-1)•(-3)•x4 + (-1)•(2)
=-18 x6 + 12x2+ 3x4 - 2
Algebraic OperationsBig IdeaPolynomials can be multiplied by polynomials:• Group variables and numerical factors• Distribute the monomial over the sum• Multiply the numerical factors• Multiply the variables
(5a -1) •(3a + 2) =(-1 + x) • (2x2 + 1) =(a + b) • (a + b) =
Algebraic Operations
HW Due Monday 11/29185.2, 185.21, 185.22, 185.24, 185.25, 185.28, 185.29, 185.32, 185.35, 185.36 and 185.43
Goal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
Algebraic OperationsBig IdeaPolynomials can be multiplied by polynomials:• Group variables and numerical factors• Distribute the monomial over the sum• Multiply the numerical factors• Multiply the variables
Algebraic Operations
(3x +1)(x2 −1) =
(3+ x)(x2 −1) =
What are the four terms if …?
Algebraic Operations
(3x +1)(x2 −1) =
(3+ x)(x2 −1) =
What are the four terms if …?
Algebraic Operations
What are the four terms if …?
5(x −2)(x−2) =5(x−2)(x+ 2) =
Try these:
4(2x +5)−3(2−7x) =5x(2 −3x) −x(3x−1) =
3a−2a(5a−a) + a2 =7x+ 3(2x−1)−8 =y(y+ 4)−y(y−3)−9y=
Try these:
(x −y)3 =(2x+1)(3x−4)(x+ 3) =
Algebraic OperationsGoal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
( y −x)3 =(y−x)(y−x)(y−x)Try this
Algebraic OperationsGoal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
( y −x)3 =(y−x)(y−x)(y−x)Try this
=(y−x)(y2 −2xy+ x2 )
Algebraic OperationsGoal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
( y −x)3 =(y−x)(y−x)(y−x)
=(y−x)(y2 −2xy+ x2 )
=y3 −2xy2 + x2y−xy2 + 2x2y−x3
=y3 −3xy2 + 3x2y−x3
So what is ( y + x)3 ?
Algebraic Operations
Big IdeaMathematical methods often work backwards from the answer to the question.
Goal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
Which of these expressions is equivalent to 121 – x2?
a. (11−x)(11+ x) b. (11−x)(11−x)
c. (x−11)(x−11) d. (x+11)(x−11)
Algebraic OperationsIf this is the answer, what is the question?
Which of these expressions is equivalent to 9x2 – 16?
a. (3x + 4)(3x−4) b. (3x+8)(3x−8)
c. (3x−4)(3x−4) d. (3x−8)(3x−8)
Algebraic OperationsIf this is the answer, what is the question?
Which of these expressions is equivalent to 9x2 – 100?
a. (9x −10)(x+10) b. (3x−100)(3x−1)
c. (3x−10)(3x+10) d. (9x−100)(x+1)
Algebraic OperationsIf this is the answer, what is the question?
Which of these expressions is equivalent to 2x2 + 10x - 12?
a. 2(x −6)(x+1) b. 2(x+ 2)(x+ 3)
c. 2(x+ 6)(x−1) d. 2(x−2)(x−3)
Algebraic OperationsIf this is the answer, what is the question?
Which of these expressions is equivalent to 3x2 - 3x - 18?
a. 3(x −3)(x+ 2) b. (3x−9)(x+ 2)
c. (3x−6)(x+ 3) d. (3x+ 6)(x−3)
Algebraic Operations
HW Due Monday 11/29185.2, 185.21, 185.22, 185.24, 185.25, 185.28, 185.29, 185.32, 185.35, 185.36 and 185.43
Goal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
HW Due Tuesday 11/30185.44, 186.46, 187.1, 187.2, any 3 of 187.3-18, 188.20, 188.22, any 3 of 188.25-32, 191.7, 191.15, and 191.
Algebraic OperationsGoal• Express and manipulate numbers using scientific notation.• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
HW Due Wednesday 12/1196.8, 196.20, 196.27, 196.28, 196.38, 196.39, 196. 45, 196. 47, 196.48, 199.2, any 4 of 199.3-26, and 199.28
Algebraic OperationsGoal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
Big Idea
xa
xb=xa−b if x≠0
For example
x5
x3=x2 if x≠0
x3
x5=x−2 =
1x2
if x≠0
Algebraic OperationsBig Idea
xa
xb=xa−b if x≠0
Do these
6x3
2x= if x≠0
5x5
2x2= if x≠0
33
2235=
Algebraic OperationsGoalExpress and manipulate numbers using scientific notation.
Big Idea
Powers of 10 can be checked by counting shifts of the decimal point that give 1.0
Algebraic Operations
Big IdeaIn simplest form the base should have a non-zero digit in the “ones place.”
Algebraic OperationsGoalExpress and manipulate numbers using scientific notation.
Write these numbers in simplest form using scientific notation:
60223000 =0.0000000315=1010 =
Algebraic Operations
Big IdeaScientific notation simplifies calculations
Examples
4500000 • 200000000 =4.5x106 • 2x108 =9x1014
0.000000003•120000 =3x10−9 •1.2x105 =3.6x10−4
Algebraic Operations
The quotient of (9.2x106) and (2.3x102) expressed in scientific notation is _______________.
What is the product of 12 and 4.2x106 expressed in scientific notation?
Big IdeaScientific notation simplifies calculations
Algebraic Operations
Do these with and without the calculator
8×10−3 • 2.25×107 =
8×10−3
2.25×107=
2.25×107
8×10−3=
13×104
=
13×10−4
=
8×10−3 • 2.25×107 =
8×10−3 •94×107 =
18×107−3 =18×104
=1.8×105
Big IdeaThe multiplicative inverse of a monomial is 1 divided by the monomial.
Algebraic Operations
For example:
6x2 •
1
6x2
⎛
⎝⎜⎞
⎠⎟=1
62 •
1
62
⎛
⎝⎜⎞
⎠⎟=62 • 6−2 =62−2 =60 =1
xa • x−a =xa−a =x0 =1 if x≠0
Algebraic OperationsBig IdeaPolynomials can be divided by monomials:• Group variables and numerical factors• Distribute the monomial over the sum• Multiply the numerical factors• Multiply the variables
For example:
(62 + 3)2
=362
+32=18 +
32=392
(6x2 + 3)2x
=6x2
2x+
32x
=3x2−1
1+32
x−1 =3x+32
x−1
Algebraic OperationsTry these:
(23 + 3)2
=
(2x3 + 3)2
=
(2x3 + 3)2x
=
4a2 −3ab+ 6a2a
=
Algebraic OperationsTry these (where x, y, z, p, a, and b ≠ 0):
(8a3 −4a2 )−4a2
+(a3 −2a2 )0.5a2
=
(y2 −5y)−y
+(2.4y5 +1.2y→ −5y)
−y=
(15z5x+ 3z2yx)3z4yx
=
14a2b2 −4ab3 + 6b6
2a2b2=
Algebraic OperationsTry these (where y and a ≠ 0):
(8a3 −4a2 )−4a2
+(a3 −2a2 )0.5a2
=
(y2 −5y)−y
+(2.4y5 +1.2y4 −0.6y3)
−0.6y2=
Algebraic OperationsGoal• Express and manipulate numbers using scientific notation.• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
HW Due Wednesday 12/1196.8, 196.20, 196.27, 196.28, 196.38, 196.39, 196.45, 196. 47, 196.48, 199.2, any 4 of 199.3-26, and 199.28
Algebraic OperationsGoal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
Homework Due Thursday 12/2201.1, 201.2, 201.3 and Test Review 6
Algebraic OperationsBig Idea
The same algorithm used to divide one number by another number can be used to divide a polynomial by a binomial.
11 121
11 because 11•11=121
11 131
11+1011
because 11•11+10=131
(a+b) (a2 + 2ab+b2 )
(a+b)
because (a+b)•(a+b)=(a2 + 2ab+b2 )
(a+b) (a2 + 2ab+b2 ) + a2
(a+b) +a2
(a+b)
because (a+b)•(a+b)+a2=(a2 + 2ab+b2 ) + a2
Algebraic OperationsBig Idea
The same algorithm used to divide one number by another number can be used to divide a polynomial by a binomial.
(a+b) (a2 + 2ab+b2 ) + a2
(a+b) +a2
(a+b)
because (a+b)•(a+b)+a2=(a2 + 2ab+b2 ) + a2
(a2 + 2ab+b2 ) + a2
(a+b)=(a+b) +
a2
(a+b)
To check multiply each term by (a +b)
Algebraic OperationsIf this is the answer, what is the question?
Which of these expressions is equivalent to 3x2 - 3x - 18?
a. 3(x −3)(x+ 2) b. (3x−9)(x+ 2)
c. (3x−6)(x+ 3) d. (3x+ 6)(x−3)
3x2 - 3x - 18
(a +b)=(c+ d) and
3x2 - 3x - 18(c+ d)
=(a+b)
Express the division of a trinomial by two different binomials using this result:
Algebraic OperationsGoal• Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.
Homework Due Thursday 12/2201.1, 201.2, 201.3, and Test Review 6
Work 201.3 by using “if this is the answer, what is the question?” thinking rather than the long division method described on page 200.
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