Unit 4 Operations & Rules. Combine Like Terms 1) 3x – 6 + 2x – 8 2) 3x – 7 + 12x + 10 Exponent...

Unit 4Unit 4Operations Operations

& Rules& Rules

Combine Like Terms

1) 3x – 6 + 2x – 8

2) 3x – 7 + 12x + 10

Exponent Rules

3) What is 2x 3x?

5x – 14

15x + 3

6x2

Warm up

Degree

The exponent for a variable

Degree of the Polynomial

Highest (largest) exponent of the polynomial

Standard Form

Terms are placed in descending order by the DEGREE

Write all answers in Standard Form!

Leading Coefficient

Once in standard form, it’s the 1st NUMBER in front of the variable (line leader)

# of Terms

Name by # of Terms

1 Monomial

2 Binomial

3 Trinomial

4+ Polynomial

Degree(largest

exponent)

Name by degree

0 Constant

1 Linear

2 Quadratic

3 Cubic

2 9y Special Names:

LinearBinomi

al

Degree Name:

# of Terms Name:

Leading Coefficient: -2

334xSpecial Names:

Cubic

Monomial

Degree Name:

# of Terms Name:

24 6x xSpecial Names:

Quadratic Binomi

al

Degree Name:

# of Terms Name:

Leading Coefficient: 4

3 27 2y y y Special Names:

CubicTrinomial

Degree Name:

# of Terms Name:

Leading Coefficient: 1

Adding Polynomial

s

2 22 4 3 5 1x x x x

1.

3x2 + x + 2

26 2 8x x 2.

x2 + 2x – 2

Subtracting

Polynomials

When SUBTRACTING polynomials

Distribute the NEGATIVE

2 23 10 8a a a a

3a2 + 10a – 8a2 + a

– 5a2 + 11a

3.

2 23 2 4 2 1x x x x

3x2 + 2x – 4 – 2x2 – x + 1

x2 + x – 3

4.

Multiplying

Polynomials

-2x(x2 – 4x + 2)

3 22 8 4x x x

5.

(x + 3) (x – 3)

2 9x

6.

(3x – 1)(2x – 4)

26 14 4 x x

7.

8. Find the area of the rectangle.

228 96 80 x x

7 10x

4 8x

9. Find the volume.

3 29 18 x x x

3x

6x

x

Warm upWarm upFind an expression for the area of

the following figure:

ChallengeChallengeFind an expression for the volume of

cylinder:

3 2( 8 12)x x x

Polynomial OpsQ8 of 20

Multiply:

2( 6)x

Find the perimeter

Find the Perimeter:

Polynomial OpsQ11 of

20

Find an expression for the volume of cylinder:


20

Write an expression for the volume of the box:


20

Find the area of the label.

Skills Skills CheckCheck

Square Roots and

Simplifying Radicals

Parts of a radicalParts of a radical

No number where the index is means it’s a square root (2)

index radicand

1. No perfect square factors other than 1 are under the radical.

2. No fractions are under the radical.

3. No radicals are in the denominator.

You try!

1.

20

52

You try!

2.

3 50

15 2

You try!

3.

120

2 30

Variables Under Square Roots

Even Exponent –

ODD Exponent –

Take HALF out (nothing left under the radical)

Leave ONE under the radical and take HALF of the rest out

15y 7y y

1026a 513a

783b 391b b

13 24a b6 12a b a

18 5 4c d

3 22 2c d c

NNthth Roots Roots & Rational & Rational ExponentsExponents

Parts of a radical

No number where the root is means it’s a square root (2)

root radicand

Simplifying Radicals

Break down the radicand in to prime factors.Bring out groups by the number of the root.

root radicand

Simplify31. 128 3 2 2 2 2 2 2 2

34 2

Simplify

x3 32. 27x x x 3 3 3 3

x3

Simplify

x4 73. 324 2 2 2 2 2 x x x x x x x

x x 4 32 2

Simplify

x934.

27x

3

3

Rewriting a Rewriting a Radical to have Radical to have

a Rational a Rational ExponentExponent

Rewriting Radicals to Rational Exponents

Power is on topRoots are in the ground

powerpower

root rootradicand radicand

Rewrite with a Rational Exponent

w5. 10 w1210


p36. 7 p137


5

7. 17 5217


y288. y28

y14


z3 69. z63

z 2

Rewriting Rational Exponents to Radicals

power power

rootrootradicand radicand


(don’t evaluate)3510. 12

35 12


(don’t evaluate)

2311. 13

23 13


(don’t evaluate)

x3212. x

3

1 i 1. 36

2. 96

3. 325

4. 8575

6i4 6i5 13i35 7i

Warm Warm upup

Powers of iand Complex Operations

““I one, I one!!”I one, I one!!”Negatives in the middle.Negatives in the middle.

1

1

i

i

1

2

3

4

i

i

i

i

75

29

251

9536

5.

6.

7.

8.

i

i

i

i

Try these!

iii

1

Add and Add and Subtract Subtract Complex Complex NumbersNumbers

Add/Subt Complex Add/Subt Complex NumbersNumbers

1.Treat the i’s like variables2.Combine the real parts then

combine the imaginary parts

3.Simplify (no powers of i higher than 1 are allowed)

4.Write your answer in standard form a + bi

Simplify

9. (3 2 ) (7 6)i i

10 8i

Simplify

10. (6 5) (1 2 )i i 6 5 1 2i i 5 7i

Simplify

11. (9 4 ) (2 3 )i i

9 4 2 3i i 7 7i

Simplify

12. 9 (10 2 ) 5i i

9 10 2 5i i 1 7i

Simplify

4 3 4 313. (11 4 ) (2 6 )i i i i 4 3 4 311 4 2 6i i i i

111 4 2 1 6i i

9 10i

Multiplying Multiplying Complex Complex NumbersNumbers

Multiplying Complex Numbers

1.Treat the i’s like variables2.Simplify all Powers of i

higher than 13.Combine like terms

4.Write your answer in standard form a + bi


14. (3 )i i 23i i

3 ( 1)i 1 3i


15. (2 3 )( 6 2 )i i 212 4 18 6i i i

12 22 6( 1)i 12 22 6i 6 22i


16. 3 8 5i i 224 15 8 5i i i

24 15 8 5 1i i 24 7 5i 29 7i

Dividing Dividing Complex Complex NumbersNumbers

What is a What is a Conjugate?Conjugate?

17. Dividing – Multiply top & bottom by the Conjugate

3 42 4

ii

2 4

2 4

i

i

2

2

6 12 8 164 8 8 16

i i ii i i

16 12 8 16

14 8 8 16

i i

i i

10 2020

i

10 2020 20

i

12

i

Unit 4 Operations & Rules. Combine Like Terms 1) 3x – 6 + 2x – 8 2) 3x – 7 + 12x + 10 Exponent...

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Transcript of Unit 4 Operations & Rules. Combine Like Terms 1) 3x – 6 + 2x – 8 2) 3x – 7 + 12x + 10 Exponent...