Algebraic Operations. Goals Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties

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


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:


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


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


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


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


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.


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


(5a -1) •(3a + 2) =(-1 + x) • (2x2 + 1) =(a + b) • (a + b) =


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




(3x +1)(x2 −1) =

(3+ x)(x2 −1) =

What are the four terms if …?


(3x +1)(x2 −1) =

(3+ x)(x2 −1) =




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) =


( y −x)3 =(y−x)(y−x)(y−x)Try this


( y −x)3 =(y−x)(y−x)(y−x)Try this

=(y−x)(y2 −2xy+ x2 )


( 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 ?


Big IdeaMathematical methods often work backwards from the answer to the question.


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)


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)


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)


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)


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


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


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


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 =


Big IdeaScientific notation simplifies calculations

Examples

4500000 • 200000000 =4.5x106 • 2x108 =9x1014

0.000000003•120000 =3x10−9 •1.2x105 =3.6x10−4


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


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.


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


Homework Due Thursday 12/2201.1, 201.2, 201.3 and Test Review 6


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


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)


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:


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.

Documents

Algebraic Operations. Goals Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties