Lesson 1.1: Product and Power Rule of Exponents

Learning Goals:

1) What are the parts of an expression?2) What is the power rule for exponents?3) What is the product rule for exponents?

Warm-Up: Match the word in column A with its best definition from column B

Answers:

1 = D 2 =A 3 = F 4 = C 5 = B 6 = G 7 = E

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Do Now: Answer the following review questions on operations with exponents.

Simplify the following examples:

1. x2 ∙ x3 ¿ x5 2. (x2 )3 ¿ x2 ∙ x2 ∙ x2= x6

Multiplication Law or Product Rule:

xa ∙ xb ¿ xa+b

The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents

Power Law or Rule for Exponents:

(xa )b ¿ xa ∙b

The “power rule” tells us that to raise a power to a power, just multiply the exponents

Examples: Multiply coefficients and add exponents!

1. 2 x2 ∙3 x4=2∙3 ∙ x2+4=¿ 2. (2 x2 y ) (7 x y3 z2 )

6 x6 14 x3 y4 z2

Power Rule: multiply exponents! distribute exponent!

3. (x3 )2 4. (−5 x y4 z2 )4

x3 ∙ x3 (−5 )4 ∙ ( x )4 ∙ ( y4 )4 ∙ ( z2 )4

x6 625 x4 y16 z8

An important part of the Common Core curriculum is paying attention to the directions!

They may ask similar looking questions, but ask for the answers in different forms.

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Practice: Rewrite each expression in the form of k xn, where k is a real number and n is an integer. Assume x≠0.

So xn means final answer, k is a real number means any #, n is an integer means a whole #, and assume x≠0 is the restriction.

5. 2 x5 ∙ x10 6. (x2 )n ∙ x3 7. (2 x3 ) (3 x5 ) (6 x )2

Product rule! Power rule! Power rule!

2 x15 x2n∙ x3 (2 x3 ) (3 x5 ) (36 x2)

Product rule! Product rule!

x2n+3 (unlike terms) (6 x8 ) (36 x2 )

216 x10

8. If x=5a4 ,∧a=2b3, express x in terms of b.

Final answer will have a b

x=5a4 substitute a=2b3 ¿5 (2b3 )4 ¿5 ∙ (24b3 ∙ 4 ) ¿5 (16b12 ) ¿80b12

9. Apply the properties of exponents to verify that the given statement is an identity. 3n+1−3n=2∙3n for integer values of n. (Show that the left side equals the right side.) cannot move terms to other side of equal sign

3n+1−3n=2∙3n

3n ∙31−3n=¿

3n (3−1 )=¿

3n ∙2=2∙3n

Homework 1.1 Product and Power Rule of Exponents3

1. Simplify: (−3 y7 z4 ) (−2 x2 y ) 2. Simplify: −(2 x4 )3

3. Simplify: (2a2b )3 ∙ (−4 ab4 ) 4. Do the following without a calculator:

Express 83 as a base of 2

5. Apply the properties of exponents to rewrite expressions in the form k xn, where n is an integer and x≠0.

a. 3 (x2 )6 (2x3 )4 b. 5 x5∙−3 x2

Lesson 1.2: Quotient, Zero, and Negative Exponent Rules4

Learning Goals:

1) What is the quotient rule for exponents?2) What is the negative exponent rule?3) What is the zero rule for exponents?

Laws of Exponents

Quotient Rule for Exponents: xa÷ xb=¿ xa−b

The Quotient Rule says that to divide two exponents with the same base, you keep the base and subtract the powers.

Examples:

a) 10a7

2a4 ¿5a3 b) 15x

5 y7 z2

3 x y2 z ¿5 x4 y5 z

Zero Rule for Exponents: x0 ¿1

The zero exponent rule basically says that any base with an exponent of zero is equal to one. Pay attention to the parentheses!

Examples:

a) 4 x0 ¿4 ∙1=4 b) (4 x)0 ¿40 ∙ x0=1 ∙1=1

Negative Exponent Rule: x−a ¿ x−a

1= 1xa or

1x−a=xa

Usually the goal is to make all exponents positive!

The negative exponent rule states that negative exponents in the numerator get moved to the denominator and become positive exponents.

The negative exponent rule also states that negative exponents in the denominator get moved to the numerator and become positive exponents.

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4 x−2

1= 4x2

or (4 x )−2=4−2 ∙ x−2

1= 142 x2

= 116 x2

Examples: Rewrite with positive exponents.

a) y−4= 1y4 b) 42a

9b3

15a12b3=14a

−3

5= 145 a3

Practice: Rewrite each expression in the form of k xn, where k is a real number, n is an integer, and x is a nonzero real number.

So xn means final answer, k is a real number means any #, n is an integer means a whole #, and x is a nonzero real number means it has an exponent.

1. 3

(x2)−3= 3x−6=3 x

6 2. x

−3 x4

x8=x−7= 1

x7

3. 3x4

(−6 x )−24. ( x2

4 x−1 )−3

3 ∙ x4

(−6 )−2 ∙ x−2 make exponents positive(x2 )−3

(4 x−1 )−3= x−6

(4 )−3 ∙ (x−1 )−3

3 ∙ (−6 )2 ∙ x4 ∙ x2 x−6

4−3 ∙ x3 make exponents (+)

108 x6 43

x6 ∙ x3=64

x9

5. Use powers of 2 to help you perform the following calculation: 27 ∙25

16

212

24=28=256

Homework 1.2: Quotient, Zero, and Negative Exponent Rules

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For questions 1 – 3: Rewrite each expression in the form k xn, where k is a real number, n is an integer, and x is a nonzero real number.

1. 3 x5

(2 x)42. 5 (x3 )−3(2 x)−4 3. x

−3 x5

3x 4

4. Express the following with positive exponents only: (5 x2 y )−2

2 x−5 y−5

5. Jonah was trying to rewrite expressions using the properties of exponents and properties of algebra for nonzero value ofx. In the given problem, he made a mistake. Explain where he made a mistake and provide a correct solution.

Jonah’s Incorrect Work: (3 x2 )−3=−9x−6

6. Rewrite using positive exponents only: 37 x7 ( y2 )−3

50 (x3 )−5

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7. If a=2b3∧b=−12

c−2, express a in terms of c.

8. Which expression is equivalent to x−1 y 4

3x−5 y−1?

(a) 13 x−4 y3 (b) 13 x

4 y5 (c) −3 x−6 y3 (4) −3 x4 y5

Lesson 1.3: Add, Subtract, and Multiply Polynomials8

Learning Goals:

1) How do we add and subtract polynomials?2) How do we multiply polynomials vertically and horizontally?3) How do we multiply polynomials with the tabular method?

Adding and Subtracting Polynomials

How do we add and subtract polynomials?

To add or subtract polynomials, add or subtract the coefficients of like terms.

If your answer is a polynomial, how should the answer be written?

Answers with polynomials should be written in standard form (highest to lowest degree)

When subtracting polynomials, what do you have to be aware of?

When subtracting, distribute the negative (or subtraction) sign, which changes each sign after the subtraction sign.

Examples:

1. (2 x3−6 x2−7 x−2 )+(x3+x2+6 x−12)

Combine like terms! 3 x3−5 x2−x−14

2. (5 x2−3 x−7 )−(x2+2 x−5)

Distribute the negative! 5 x2−3 x−7−x2−2x+5=4 x2−5 x−2

3. (x3+2 x2−3 x−1 )+(4− x−x3)

Combine like terms! 2 x2−4 x+3

4. (x2−3 x+2 )−(2−x+2 x2)

Distribute the negative! x2−3 x+2−2+x−2 x2=−x2−2 x

VERTICAL example:

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a2+a−3¿2a2+3a−1

−1(a2+a−3)= −a2−a+33a(a2+a−3)= 3a3+3a2−9a2a2(a2+a−3)=2a4+2a3−6 a2

=2a4+5 a3−4 a2−10a+3

HORIZONTAL example:(2a+1)(a2−2a−2)=2a3−4a2−4a+a2−2a−2=2a3−3a2−6a−2

a. Multiply −x2+2x+4∧x−3 in a horizontal format.

(−x2+2x+4)(x−3) Distribution!

−x3+3x2+2 x2−6 x+4 x−12 Combine like terms!

−x3+5x2−2x−12

b. Multiply x−1 , x+4 ,∧x+5 in a horizontal format.

(x−1)(x+4)(x+5) Distribute!

x2+4 x−x−4 Combine like terms!

(x2+3 x−4 ) ( x+5 )

x3+5x2+3 x2+15 x−4 x−20

x3+8 x2+11 x−20

Multiplication of Polynomials in a Tabular Format (Area Arrays)

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Example: Use tabular method to multiply (x+8)(x+7) and combine like terms

Add all boxes:

( x+8 ) ( x+7 )=x2+15 x+56

Use the tabular method to multiply

(x2+3 x+1)(x2−2)

Add all boxes: x4+3 x3−x2−6 x−2

b) Use the tabular method to multiply (x2+3 x+1 ) (x2−5 x+2 ) and combine like terms.

x2 −5 x +2x4 −5 x3 2 x2 x2

3 x3 −15 x2 6 x +3 xx2 5 x 2 +1

Add along the diagonal! x4−2x3−12 x2+x+2

Practice:

Multiply the following expressions and express each product in simplest form.

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1. 2a(5+4a) 2. (2 r+1)(2r 2+1)

10a+8 a2=8a2+10a 4 r3+2 r+2r 2+1=4 r3+2r2+2 r+1

3. (x+5)(x2+3 x+2) 4. (2 x+5 )3

x3+3x2+2 x+5 x2+15 x+10 write 2 x+5 three times and then multiply!

x3+8 x2+17 x+10 (2 x+5)(2 x+5)(2 x+5)

(4 x2+10 x+10 x+25 ) (2 x+5 )

(4 x2+20 x+25 ) (2 x+5 )

8 x3+40 x2+50 x+20 x2+100 x+125

8 x3+60x2+150 x+125

You cannot use the power law on polynomials!

(2 x )2=23 x2=4 x2 (x+2)2≠ x2+4

5. Use the tabular method: (−x2−4 x+4)(x+3)

−x2 −4 x 4−x3 −4 x2 4 x x−3 x2 −12 x 12 +3

Add along the diagonal! −x3−7 x2−8 x+12

Applications of Multiplying Polynomials:

6. Write an expression for the volume of a rectangular prism with length of

2 x+2, width of x+1, and height of x+3.

B = area of base (on reference sheet) General Prisms: V=Bh

h = height Base is a rectangle so A=l x w

V=Bh

V=l ×w×h

V=(2 x+2)(x+1)( x+3)

7. Write an expression for the volume of a cylinder with radius of x−2 and height of 3 x−4.

(on reference sheet) Cylinder: V=π r2h12

V=π ( x−2 )2(3 x−4)

V=π ( x−2 ) ( x−2 ) (3x−4 )

V=π (x2−4 x+4 )(3 x−4 )

V=π (3x3−12 x2+12x−4 x2+16 x−16)

V=π (3x3−16 x2+28 x−16)

Homework 1.3 Adding, Subtracting, and Multiplying Polynomials

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1. ( x+1 )−( x−2 )−(x−3) 2. (2 x3−x2−9 x+7 )+(11 x2−6 x3+2 x−9)

3. Find the product of (3 x+2 )∧(3x+5) in simplest form using the vertical format.

4. Find the product of (5 x3+4 )2 in simplest form using the horizontal format.

5. Find the product of the binomial (4 x+3) with the trinomial (2 x2−5x−3) in simplest form using an area array.

6. Find the volume of a rectangular prism if the length is(x−3), the width is

(x+2), and the height is (x+4 ) in simplest form.

Lesson 1.4: Use Structure to Prove-Find Pythagorean Triples

Learning Goals:

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1. How do we prove Pythagorean Triples?2. How do we find Pythagorean Triples?3. How do we prove a Pythagorean Identity?

REVIEW:

a) What is the Pythagorean Theorem formula and when can you use it?Pythagorean Theorem formula is on the reference sheet. Use it when you have a missing side of a right triangle. a2+b2=c2

b) Find the missing side of the triangle given below:c is the longest side

a2+b2=c2

52+122=x2

25+144=x2

169=x2

13=x

c) Prove that the 3 sides of the triangle above are a Pythagorean triple.{5, 12, 13}a2+b2=c2

52+122=132 goal is to get left side = right side of equation25+144=169

169=169 √

Advice for Proving or Showing Identity Equations:

Your end goal is to get the left side = right side Choose a side of an equation to begin working on Sometimes you might have to work on both sides When “proving” or “showing”, you are NOT allowed to cross the equal sign!

Example 1: Show how “the polynomial identity”

(x2+ y2 )2=(x2− y2 )2+(2 xy )2 can be used to generate Pythagorean triples.”

This can be difficult to do at first. Let’s first prove the identity when y=1.15

Prove that if x>1, then a triangle with side lengths x2−1.2x .∧x2+1 is a right triangle.

x2+1is thelongest side

a2+b2=c2

(x2−1)2+(2 x)2=(x2+1 )2 (easiest to work with both sides)

(x2−1 ) (x2−1 )+2 x2=(x2+1 )2

(x¿¿4−x2−x2+1)+(22 ∙ x2 )=(x2+1 ) (x2+1 )¿

(x¿¿4−2 x2+1)+(4 x2)=x4+x2+ x2+1¿

x4+2 x2+1=x4+2x2+1

Example 2: Jeremy uses the polynomial identity

(x2− y2 )2+(2 xy )2=(x2+ y2 )2 to generate the Pythagorean Triple 9 ,40 ,41.What values of x∧ y did he use to generate the values for the three sides of a right triangle?

(1) x=4 , y=3 (2) x=40 , y=9 (3) x=5 , y=4 (4) x=25 , y=16

Method 1: substitute multiple choice answers in!

Method 2: composing the 2 formulas!

a2+b2=c2

(x2− y2 )2+(2 xy )2=(x2+ y2 )2 a=x2− y2b=2 xy c=x2+ y2

Pick the easiest to work with, which is b=2 xy

{9,40,41 }a ,b , c

40=2 xy 20=xy

Proving Polynomial Identities: Show that each of the following is a polynomial identity:

1. Show that (x2+ y2 )2=(x2− y2 )2+(2 xy )2 for all real numbers x∧ y.

Make both sides equal!

(x2+ y2 )2=(x2− y2 )2+(2 xy )2 (easiest to work with both sides)

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(x2+ y2 )( x2+ y2 )=(x2− y2 )(x2− y2)+(22x2 y2) Distribute and power law!

x4+x2 y2+x2 y2+ y 4=x4−x2 y2+ y4+4 x2 y2

x4+2 x2 y2+ y4=(x4−2 x2 y2+ y4 )+4 x2 y2

x4+2 x2 y2+ y4=x4+2x2 y2+ y4√

2: Show that (9, 12, 15) is a Pythagorean triple.

a2+b2=c2

92+122=152

81+144=225

225=225√

3. Prove: (a+b)2−(a−b )2=4 ab (easiest to leave one side alone)

(a+b ) (a+b )−¿

a2+2ab+b2−(a2−2ab+b2 )=¿

a2+2ab+b2−a2+2ab−b2=¿

4 ab=4 ab√

Homework 1.4 Use Structure to Prove/Find Pythagorean Triples

1. Show that (12 ,35 ,37) is a Pythagorean triple.

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2. Show that a3−b3=(a−b)(a2+ab+b2) for all real numbers a∧b.

3. Sam and Jill decide to explore a city. Both begin their walk from the same starting point.

Sam walks 1 block north, 1 block east, 3 blocks north, and 3 blocks west.

Jill walks 4 blocks south, 1 block west, 1 block north, and 4 blocks east.

If all the city blocks are the same length, who is the furthest from the starting point?

Lesson 1.5: DIVISION OF POLYNOMIALS (Day 1)

Learning Goals:

1. What is the quotient rule for exponents?2. How do we divide polynomials using long division?

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Division of a Polynomial by a Monomial

What is the rule needed to solve the problems below?

When you divide like bases, you can subtract the exponents!

When you divide by a monomial, separate into multiple fractions!

1. 15 y5−20 y4

−5 y2 2. 18x

3 y5+36x3 y4−6 x2 y2

−6 x2 y2

15 y5

−5 y2+−20 y4

−5 y2 18x3 y5

−6 x2 y2+ 36 x

3 y4

−6 x2 y2+−6 x2 y2

−6 x2 y2

−3 y3+4 y2 −3 x y3−6 x y2+1

Dividing a Polynomial by a Polynomial using the Reverse Tabular Method for Multiplication.

1. Reverse the tabular method of multiplication to find the quotient: 2x2+x−10x−2

So you can work backwards to fill in the missing boxes.

What is the quotient: 2 x+5

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2. Reverse the tabular method of multiplication to find the quotient: 2x3+15 x2+27 x+5

2 x+5

What is the quotient: x2+5x+1

Can you use the reverse tabular method for multiplication if there is a remainder? NO

LONG DIVISION of POLYNOMIALS focus on the highest power

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Example:

¿¿ ¿ 0 ¿¿ ¿ ¿ 2 x2+7 x ¿ 2 x2+6 x ¿ x+3 ¿ x+3 ¿ 0 ¿ ¿3. Use long division to find the following: 2x

2+x−10x−2

2 x+5

x−2|2x2+x−10

−(2 x2−4 x )

5 x−10

−(5x−10)

0

4. Using long division, divide x2+3x+5by x+1

x+2

x+1|x2+3 x+5

−(x2+x ) x+2+ 3x+1

2 x+5

−(2x+2)

3 is the remainder

a. What happens in this problem that we have not seen before?

there is a remainder of 3!

b. What does this mean? x+1 is not a factor!

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2x2+x−10x−2

=2 x+5 x2+3 x+5x+1

=x+2+ 3x+1

2 x2+x−10=(2x+5)(x−2) x2+3x+5=( x+2 ) ( x+1 )+3

5. Is x2+1 a factor of −4 x3+3 x4+5+12 x2? Justify your answer.

Dividing by a polynomial, must use long division

Reorder! 3x4−4 x3+12 x2+0x+5

x2++0 x+1Missing the x, so we must add it in!

3 x2−4 x+9

x2+0x+1|3 x4−4 x3+12x2+0 x+5

−(3x4+0 x3+3 x2)

−4 x3+9x2+0 x

−(−4 x3+0 x2−4 x)

9 x2+4 x+5

−(9 x2+0 x+9)

4 x−4

3 x2−4 x+9+ 4 x−4x2+1

x2+1 is not a factor because the remainder is

not zero

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Division Algorithm:If f ( x )∧d (x) are polynomials such that d (x )≠0, and the degree of d (x ) is less than or equal to the degree of f (x), there exist unique polynomials q ( x )∧r (x) such that:

f ( x )=d ( x ) ∙q (x )+r ( x)

Where r ( x )=0 or the degree of r (x ) is less than the degree of d (x ). If the remainder equals zero, d (x ) is a factor of f (x).

6. The expression x2+5 x−20x−3

is equivalent to

(1) x+8+ 4x−3 (2) x+8− 4

x−3 (3) x+8 (4) x−8+ 4x−3

Use long division because you are dividing by a polynomial.

x+8

x−3|x2+5x−20

−(x2−3x )

8 x−20

−(8 x−24)

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Step 1:

Make sure the polynomial is written in descending order. If any terms are missing, use a zero to fill in the missing term (this will help with the spacing).

Step 2:

Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol.

Step 3:

Multiply (or distribute) the answer obtained in the previous step by the polynomial in front of the division symbol.

Step 4:

Subtract and bring down the term.

Step 5:

Repeat Steps 2, 3, and 4 until there are no more terms to bring down.

Step 6:

Write the final answer. The term remaining after the last subtract step is the remainder and must be written as a fraction in the final answer.

Name: ____________________________________23

Homework 1.5 Division of Polynomials

1. 15x5 y4−25x3 y3−5 xy

−5 xy 2. 36 x2 y4+42x5 y3 z2

−6 x2 y

3. Divide using long division: 4 x2−4 x−352 x−7

4. Divide using reverse tabular method for multiplication: x3+2 x 2+2 x+1

x+1

5. Given a ( x )=−8x4−10 x3+17 x2−4∧b ( x )=4 x2−5x+2.

Express a(x )b(x ) in the form q ( x )+ r (x)

b(x )where r (x )<b (x).

Homework 1.6: Long Division24

Directions: Find the quotient using long division.

1. (x+8+6 x3+10 x2 )÷ (2 x2+1 ) 2. (1+3 x2+x 4 )÷ (3−2 x+x2 )

3. Divide 8 x4−5by2 x+1 4. (x3−9 )÷ (x2+1 )

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