Unit 1 Notes

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Unit 1 Notes

• Conversions, Polynomials, and Solving Equations

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Interpreting Terms, Factors, and Coefficients
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Vocabulary

• A variable is a letter or symbol used to
  represent the unknown
• Example x, y, A, V
• A constant is a value that does not change can
  be positive or negative
• Example 4, -32, pi

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• A numerical expression may only contain constants
  and/or operations
• Example 4 3, 100 10
• An algebraic expression may contain variables,
  constants, and/or operations
• Example x 4, x2 - 9

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• Factors are items that are being multiplied
  together
• Examples 7(5), 4(x 2)
• An equation is an expression with an equal sign
  ().
• Example 4x 8

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• Terms refers to how many items are being added,
  subtracted, or divided
• Example x2 x - 2 has three terms (x2, x, -2)
• Like terms have the same variables raised to the
  same power.
• Example x2 and 5x2 are like terms
• Example x2 and x are not like terms
• Example -3x2 4x2 6x 10x 5 - 4

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• A coefficient is the number in front of the
  variable.
• The base is the part of the expression that has
  been raised to a power.
• The exponent is the number being raised up next
  to the base also described as the number of
  times you multiply something by itself
• 4x 102 7x2

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• The leading coefficient is the coefficient of the
  term with the largest degree.
• Example 3x2 5
• Example 6x 3 12x2
• Example -2x 4x3 3

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• Classifying Expressions by Degree

Degree Example Name
0 5x0 or 5 Constant
1 7x1 or 7x Linear
2 10x2 7x Quadratic
3 6x3 - 10x2 7x Cubic
4 8x4 5 Quartic
5 3x5 -6x3 5 Quintic
gt 5 7x7 4x2 92 Polynomial w/ degree
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• Classifying Expressions by Terms
• Standard Form putting the terms in order of
  highest degree to lowest degree

of Terms Example Name
1 5, 7x, 2x2 Mononomial
2 3x 7 Binomial
3 5x2 2x 8 Trinomial
gt 3 8x4 - 3x2 4x - 9 Polynomial w/ terms
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• Example 9x 7 2x2
• Rewrite the above expression in standard form.
• Identify the coefficients of the expression.

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• Example 9x 7 2x2
• Identify the expression by terms. (How many terms
  are there?)
• Identify the expression by degree. (What is the
  highest degree in the expression?)

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• Example 2x4y3
• Identify all the exponents.
• Identify all the factors.

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• Simplify the expression by combining like terms
• 8x2 12x2 - 9y 7y 45 - 10

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Properties of Equality
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• Properties of Operations
• Associative Property of Addition Regardless of
  the grouping, the sum of multiple addends will be
  the same.
• (a b) c a (b c)
• Commutative Property of Addition Regardless of
  the order, the sum of multiple addends will be
  the same.
• a b b a
• Additive Identity Property of 0 The sum of any
  number and zero is that number.
• a 0 a
• Additive Inverse The inverse of a is a, so that
  their sum is zero.
• a (-a) 0

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• Properties of Operations
• Associative Property of Multiplication
  Regardless of the grouping of the factors, the
  product will be the same.
• (ab)c a(bc)
• Commutative Property of Multiplication
  Regardless of the order of the factors, the
  product will be the same.
• ab ba
• Multiplicative Identity of 1 The product of any
  number and 1 is that number.
• a 1 a
• Multiplicative Inverse The inverse of a is
  (1/a), so that the product is one.
• a (1/a) 1

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• Properties of Operations
• Distributive Property The sum of two numbers (a
  and b) multiplied by a third number (c) is equal
  to the sum of each addend (a and b) multiplied by
  the third number (c).
• c(a b) ac bc

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• Name the property for each of the following
• 5 6 6 5
• 3(5 6) (3 5) (3 6)
• 4 (7 8) (4 7) 8
• 78 0 78
• 6 (1/6) 1

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• Properties of Equality
• Reflexive Property Anything is equal to itself.
• a a
• Symmetric Property If a b, then b a.
• Example 2x 4 is the same as 4 2x
• Substitution Property If a b, and b equals
  some quantity, then it can be substituted for b.
• Example If a b, and b 5, then a 5.
• Transitive Property Using the Substitution
  Property
• If a b, and b c, then a c.

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• Properties of Equality
• Addition Property of Equality If you add the
  same number to both sides of an equation, the
  equation is true.
• If a b, then a c b c.
• Subtraction Property of Equality If you subtract
  the same number to both sides of an equation, the
  equation is true.
• If a b, then a - c b - c.
• Multiplication Property of Equality If you
  multiply the same number to both sides of an
  equation, the equation is true.
• If a b, then a c b c.
• Division Property of Equality If you divide the
  same number to both sides of an equation, the
  equation is true.
• If a b, then a/c b/c.

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• Name the property for each of the following
• If a b, then a 5 b 5.
• If a b, then a/6 b/5.
• If a b, and b 7, then a 7.
• If 5x 4 10, then 10 5x 4.

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Solving 1 and 2-step Equations
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• An equation is a mathematical statement that sets
  two expressions equal to each other.
• Example 4x 8
• The solution to an equation is a value that makes
  the equation true.
• Example If 4x 8, then x 2 because
• 4x 8
• 4(2) 8
• 8 8 a

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• How do we find the solution of an equation?
• You can rearrange the equation to isolate the
  variable by using inverse (opposite) operations.
  This is called solving for a variable.

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An equation is like a scale. To make sure the
scale is balanced, perform the same (inverse)
operation to both sides.

• Given
• Divide both sides by 4
• (Division Property of Equality!)
• Done! ?

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• Solve for x 5 8x 3
• Operations Properties

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• Solve for x 2x 10 x - 5
• Operations Properties

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Rearranging Formulas
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• Now, what if we had an equation that only
  contained variables no constants whatsoever?
  For example, a b c.
• What would we do?
• Just use inverse (opposite) operations to solve
  for the variable specified!

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Writing and Solving Equations
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• Did you know math has its own language (e.g
  terms, exponents, coefficients)?
• What if we didnt have this language? How would
  we get from English phrases back to numbers and
  variables?

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• Each operation (addition, subtraction,
  multiplication, and division) has its own key
  words that hint at the operation.

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Addition

• Sum
• Plus
• Add
• More than (Hint add and switch)
• Increased by
• Together

• Examples
• The sum of a number and 7
• x 7
• 5 more than a number
• x 5
• A number plus 8
• x 8

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Subtraction

• Difference
• Minus
• Decreased
• Less than (Hint switch the order)
• Take Away

• Examples
• The difference of some number, x, and 7
• x 7
• Three less than a number
• x - 3

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Multiplication

• Product
• Twice
• Double
• Triple
• Times

• Examples
• Product of 5 and 7
• 5 7
• Twice and Double
• 2 x or 2x
• Triple
• 3 x or 3x

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Division

• Quotient
• Half
• Divide by

• Examples
• The quotient of a and b
• A number divided by 9

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Exponents

• Square
• Cubed
• To the power of
• Raised to a power

• Examples
• A number squared
• x2
• A number to the power of 5
• x5

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1. The sum of a number and 10.
2. The product of 9 and x.
3. 7 less than g.
4. The product of 5 and x squared.
5. 9 less than j to the fourth power

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Lets try writing numerical expressions as verbal
translations, or in words.

1. x 3
2. m 7
3. 2y
4. k 5
5. 8 3x
6. x5

• 7.

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1. Eve reads 25 pages per hour. Write an expression
   for the number of pages she reads in h hours.
2. Sam is 2 years younger than Sue, who is y years
   old. Write an expression for Sams age.
3. William runs a mile in 12 minutes. Write an
   expression for the number of miles that William
   runs in m minutes.

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Writing, Solving, and Graphing Inequalities
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Whats an inequality?

• A range of values rather than ONE set number or
  answer
• An algebraic relation showing that a quantity is
  greater than or less than another quantity.
• Speed limit

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Symbols
Less than
Greater than
Less than OR EQUAL TO
Greater than OR EQUAL TO
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Solutions.
You can have a range of answers
All real numbers less than 2 x lt 2
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Solutions continued
All real numbers greater than -2
x gt -2
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Solutions continued.
All real numbers less than or equal to 1
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Solutions continued
All real numbers greater than or equal to -3
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Did you notice
Some of the dots were solid and some were open?
Why do you think that is?
If the symbol is gt or lt then dot is open because
it can not be equal. If the symbol is ? or ? then
the dot is solid, because it can be that point
too.
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Solving an Inequality
Solving a linear inequality in one variable is
much like solving a linear equation in one
variable. Isolate the variable on one side using
inverse operations.
Solve using addition
x 3 lt 5
Add the same number to EACH side.
x lt 8
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Solving Using Subtraction
Subtract the same number from EACH side.
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Using Subtraction
Graph the solution.
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Using Addition
Graph the solution.
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THE TRAP..
When you multiply or divide each side of an
inequality by a negative number, you must reverse
the inequality symbol to maintain a true
statement.
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Solving using Multiplication
Multiply each side by the same positive number.
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Solving Using Division
Divide each side by the same positive number.
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Solving by multiplication of a negative
Multiply each side by the same negative number
and REVERSE the inequality symbol.
Multiply by (-1).
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Solving by dividing by a negative
Divide each side by the same negative number and
reverse the inequality symbol.
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Some things to remember!
When you multiply or divide by a negative number
on both sides of the inequality, the inequality
sign changes or "flips."
HINT When graphing inequalities, always keep the
variable on the LEFT HAND SIDE. The inequality
sign will tell you in which direction to shade.
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Operations with Polynomials
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• Degree of a Term the value of the exponent of
  the variable
• Example What is the degree of ?
• Degree of a Polynomial the largest degree of its
  terms
• Example What is the degree of the polynomial
  ?

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• Classifying Expressions by Terms

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• Example
• Write the following polynomial in standard form.
  
• State its degree and leading coefficient.
• Classify by the degree and number of terms.

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Remember!

• When combining like terms
• Combine the coefficients
• The exponents do not change!

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• Simplify.

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Addition Vertical Method

• When adding two numbers using the vertical
  method, we line up the place values vertically
• 463
• 239
• 702

Hundreds Tens Ones
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Addition Vertical Method

• Similarly, we put the polynomials in standard
  form and line up the like terms of each
  polynomial.
• Example

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Addition Horizontal Method

• When using the horizontal method, identify the
  like terms and add them.
• Example

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Subtraction Vertical Method

• To subtract, write the polynomials in standard
  form and line up the like terms. Then, add the
  opposite.
• Example

?
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Subtraction Horizontal Method

• To subtract, find the like terms and add the
  opposite.
• Example

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• Simplify the expression.

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• Simplify the expression.

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Multiplying Polynomials

• Remember when multiplying, you are using the
  Distributive Property!
• So
• Multiply the coefficients
• Add the exponents
• Combine like terms

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• Simplify the expression.

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• Simplify the expression.

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• Simplify the expression.

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Multiplying Binomials

• When multiplying two binomials, you will have a
  total of four terms.
• Example
• In order to carry out this operation we must FOIL
  (again, the Distributive Property!).

1st 2nd 3rd 4th
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• FOIL stands for
• First multiply the first and third terms in each
  set of parentheses
• Outer multiply the first and last terms
• Inner multiply the second and third terms
• Last multiply the second and fourth terms

• Example (2x 5)(3x -3)
• First 2x and 3x
• 2x(3x) 6x2
• Outer 2x and -3
• -3(2x) -6x
• Inner 5 and 3x
• 5(3x) 15x
• Last 5 and -3
• -3(5) -15

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Next, we combine like terms 6x2 6x 15x
15 6x2 9x 15 Done! ?
Example (2x 5)(3x -3) First 2x and
3x 2x(3x) 6x2 Outer 2x and -3 -3(2x)
-6x Inner 5 and 3x 5(3x) 15x Last 5 and
-3 -3(5) -15

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• Multiply

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Dividing Polynomials by Monomials

• Divide each term in the dividend by the divisor
• Simplify each fraction using this rule

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• Divide

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• Divide

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Factoring Polynomials

• Find two numbers, m and n, such that
• m n b
• m n c
• The solution will be in the form

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• Factor

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• Factor