Alg1 Chapter 4: Polynomials

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Alg1 Chapter 4 Polynomials

• Addition and Subtraction
• Multiplication
• Problem Solving

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4-1 Exponents

• Objective Write simplify expressions
  involving exponents

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4-1 Exponents Definitions

• Powers
• 5 to the 0 power 50 1 anything raised to 0
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• 1st power of 5 51 5 read as five to the
  1st power
• 2nd power of 5 52 55 five to the 2nd power
  or 5 squared
• 3rd power of 5 53 555 five to the 3rd
  power or 5 cubed
• 4th power of 5 54 5555 five to the 4th
  power
• The expression bn
• tells you that the base b is used as a factor n
  times
• Given 54 base 5 exponent 4

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4-1 Exponents x2 different from 2x
Very different from X X 2X X X X 3X X
X X X 4X x² 2x² 3x² Where as x2
x is simplified (cannot add not like terms)

• Powers
• xx1
• xxx2
• xxx x3
• xxxx x4
• 2xxx2 x3
• Also
• x2 x x x x x3

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4-1 Exponents Examples
Write using exponents or in exponential
form A) 3 3 3 3 B) a a
bbbcccc C) -2 p q 3 p q q p
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4-1 Exponents Order of Operations

• Remember PEMDAS (grouping sym.s), exponents,
  mult/div. in order, and add/sub in order
• Parenthesis Very Important - Caution when
  exponent outside ( )
• BEWARE (-2)² (-2)(-2) 4 versus
  -2² - 22 -4
• neg. s raised to an even exponent are
  positive
• neg. s raised to an odd exponent are
  negative
• (2y)3 (2y) (2y) (2y) 8y3
• 2(y)3 2y3
• 3 (X)2 3 x²
• (3X) 2 9 x²
• Evaluate x4 if x -5,
• use ( ) to hold x spot -------- very important
  !!!!
• (-5) 4 (-5) (-5) (-5) (-5) 625
• NOT -5 4 which is - 5 5 5 5 - 625

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4-1 Exponents Examples

• Simplify a. -34 b. (-3) 4 c. (15) 2
  d. 1 52
• Evaluate (2a b) 2 if a 3 and b -2
• Evaluate if x 3 and y -2
• 3x
• Find the area of the rectangle
• x
• Square cube
• L w L w h
• Assign p. 143 w 1-31 odd 33-45 m3

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L w
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Warm up before 4-2

• Write - 4rs2rsr in exponential form
  5y
• Find the area of the rectangle 2y
• Evaluate each expression if m -2.
• 4m3
• (4m) 3
• Simplify
• -32
• (-3) -32
• (-3) 2
• (7 - 3) 2
• 7 32
• (7 - 3) 2
• 7 32
• Evaluate (3r 2s)2 if r 2 and s -4

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Warm up before 4-2

• Write - 4rs2rsr in exponential
  form -8r³s² 5y
• Find the area of the rectangle 2y 10y²
• Evaluate each expression if m -2.
• 4m3 -32
• (4m) 3 -512
• Simplify
• -32 -9
• (-3) 2 9
• (7 - 3) 2 16
• 7 32 -2
• Evaluate (3r 2s)2 if r 2 and s -4
  4

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4-2 Adding Subtracting Polynomials
Objective to add and subtract polynomials
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4-2 Adding Subtracting Polynomials
When you read a sentence, it is split up into
words. There is a space between each
word. Likewise, a mathematical expression is
split up into terms by the /- A
term is a number, a variable, or a product or
quotient of numbers and variables raised to
powers.
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4-2 Adding Subtracting Polynomials Terms
Monomials (polynomial of 1 term a one word
math expression)
14 constant
-6x2y (-6 is the coefficient)
r (2/3 is called the coefficient)
Z (1 is the Coefficient)
Polynomials (sum or difference of
monomials which are terms)
14 z Binomial (2 terms)
-6x2y x 14 Trinomials (3 terms)
To Add or Subtract terms must be similar must
have like terms Two monomials are similar if
have the same variables to the same powers -
their coefficients can be different.
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4-2 Adding Subtracting Polynomials
Not Polynomials has division
has abs. value

• Polynomials
• X
• 2x 10
• 2x³ y² x²

Polynomials are expressions that have no
operations other than addition, subtraction, and
multiplication by or of the variables.
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4-2 Adding Subtracting Polynomials Examples
Similar -5xy2 , 16yxy, xy², 2

NOT Similar -2xy² , -2x²y

• -3x² 5x -2x x² - 4
• 4m² 5mn² - m² 3mn²
• Write the sum of the areas of the rectangles in
  simplest form

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4-2 Adding Subtracting Polynomials Degrees

• Degree of a variable the number of times the
  variable occurs in the monomial (when
  simplified its the exponent of the variable)
• Degree of a monomial (term) sum of degrees of
  its variables-all of them
• Degree of polynomial greatest of the degrees of
  its terms (after simplification)
• Examples
• 5x²yz4 degree of x ___________
• degree of y ____________
• degree of z ____________
• degree of term (monomial)__________
• 5x²yz4 xyxyxzzz degree of polynomial
  ____________

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4-2 Adding Subtracting Polynomials Examples

• Simplify -6x³ 3x² x² 6x³ - 5
• Find the degree of 4x³yz²
• Add 2a² ab 2b and 4a² - 3ab 9
• Subtract 2x² - y² from 5x² 7xy 2y²
• Find four consecutive integers whose sum is twice
  the cube of 5
• p.143mixed review 1-9 p.148w3-48m3

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Match Terms
a. In 43, the 3 is b. In 43 , the 4 is c. for
3 3 3 3, it is 4th d. 3 2 6 e. 3
2 f. (a 3)/4 g. a, x or y h. lt, gt i. The 7
in 35 / 5 7 j. The 2 in 7 5 2 k. -2 gt -5
l. The 6 in 2 3 6
Numerical expression Equation Inequality
symbol Inequality Difference Product Quotient Powe
r Base Exponent Variable Algebraic expression
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4-3 Multiplying Monomials Rules

• Objective To multiply monomials

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Rules of Exponents

• Zero power (4bc)0 1
• Negative exponents 2x-1 2 ( )
• Multiply (product rule)
• Power rule (2²)³ 223 64
• Division rule
• x³
• Product power
• Quotient power

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Simplify
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• True or False

1. 2. 3. 4.
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4-3 Examples w/ Surface Area

• S A Add the area of each face
• p.150 problems 2-14even p.153w3-33m3

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4-4 Powers of Monomials

• Objective To find powers of monomials.
• Apply the Power Rule -
• the Product Power Rule -

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4-4 Powers of Monomials

• Power Rule
• Simplify
• (u4)5
• (-a)²3
• (-2k)5
• (-3x²y5 )³

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4-4 Powers of Monomials

• Evaluate if t 2
• 3t³
• (3t)³
• 3³t³
• p.153w36,38,41,42 p.156 w 1-20

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4-5 Multiplying Polynomials by Monomials

• Objective to multiply a polynomial by
  a monomial

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4-5 Multiplying Polynomials by Monomials

• Given x (x3) use distributive property
• rules of exponents
• Examples
• X (x3) x² 3x
• -2x ( 4x² - 3x 5) -8x³ 6x² - 10x
• 5xy² (3x² - 4xy y²) 15x³y² - 20x²y³ 5xy4
• Solve n(2 5n) 5(n²-2) 0 n 5
• p. 156w24-51m3 p.159w3-39m3,40-44

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4-5 Multiplying Polynomials by Monomials

• More examples
• Multiply
• 6(x2)
• -4(y-3)
• 2a²b ab² - ab
• -ab
• 4y(y²-2y 1)
• 3pq²(2p²q pq 5q²)

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4-6 Multiplying Polynomials
Objective Multiply 2 Polynomials
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4-6 Multiplying Polynomials
may hear this called FOIL for first, outer,
inner, last which only works for multiplying 2
binomials Well use a method that works for
all..
Multiply (2x 5)(3x 2)
Multiply 2x² - 5x 4 3x
- 2
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4-6 Multiplying Polynomials

• Day 1 p.162-64w3-42m3 p.160mr
• Day 2 Worksheet for grade front/back evens

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Rules About Exponents
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Rules About Exponents
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4-7 Transforming Formulas Literals

• Objective Rearrange a formula to solve for a
  particular variable

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4-7 Transforming Formulas Literals

• Solve for the variable in red
• As²2rs
• P 0.7854d²sn
• p.166w2-26even p.163-4 w 43-50a

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Solving Word Problems Step 1 Identify the
formula, for example D r t Step 2
Sketch to help visualize the situation Step 3
Chart to organize information list
subjects down write formula across fill in
what you know (check for same units) Step
4 Set up Equation (must have ) Step 5 Solve
then answer what the problem asks
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p.170p1-15 odd p.166 mixed review
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• Day 2 worksheet after 4.8 Ch Rvw

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4-9 Area Problems
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4-9 Area Problems
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4-9 Area Problems
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4-10 Problems w/out Solns
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p.173/4 problems 2-14 evens mixed review p.176
problems 2-8 even
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• p.177 self test 3
• p.180 Chp Test all