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Welcome to Unit 1
Our Topics for this week
• Welcome and Syllabus Review
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Brief Syllabus Review
Contact Information for Instructor
Seminar Rules
Discussion
• Topics
– Polynomials
• Addition of Polynomial
• Subtraction of Polynomials
• Evaluating a Polynomial
– Multiplying Polynomials
Syllabus Highlights
• Due Dates
All learning activities for a unit are due by Tuesday
11:59 pm ET.
• Instructor Contact
Email: [email protected]
AIM name: kathrynbaalman
Download AIM from
http://www.aim.com/get_aim/win/other_win.adp
Seminar Rules, Structure
• Usual format
– Discussion of a concept
– Responses to questions I post on the concepts
• Posting a wrong answer will not negatively affect your
participation grade.
• It is important that you try to participate rather than
simply watch
• Do not interrupt if you enter seminar late
• Social posts (It is raining, Hi, Bye, My car broke down)
are not appropriate and will not help your participation
grade.
Replying to Instructor
When I want each of you to reply to a question or solve
a problem, I will say:
EVERYONE: followed by my question.
If no specific answer is requested, your response will be
made by either typing Yes or No, followed by the
Enter key, or clicking Send.
Polynomials
A polynomial is a sum of terms of the
form axn, where a may be any real
number and n a whole number.
– The power n may be ZERO, so a constant
is a polynomial
– Terms are written in descending order of
the powers
EXAMPLE: 3x3 + 5x2 – 8x + 7
Polynomials
Polynomials are identified by the number
of terms they contain:
• One term – a MONOMIAL
• Two terms – a BINOMIAL
• Three terms – a TRINOMIAL
• Four or more terms – a POLYNOMIAL
DEGREE of Terms, Polynomials
The degree of a term is the sum of the
powers that appear in all of the
variables in the term.
EXAMPLE:
x2y2 is of degree 4
The degree of a polynomial is the
highest power that occurs in the
polynomial.
EXAMPLE:
x2 + 2x + 5 is of degree 2
Polynomials
Certain situations cause an expression to
NOT be a polynomial. These are:
• There’s a variable in a denominator, 2/x
• There’s a variable under a radical, √x
• There’s a number or variable with a
fractional or negative exponent, x⅓
Adding Polynomials
Example:
(9x3+ 6x – 8) + (-3x2 – x – 5)
= 9x3 + 6x – 8 - 3x2 – x – 5 [Remove
parentheses. Since it is an addition sign between
parentheses that PLUS sign is dropped]
= 9x3 - 3x2 + 6x – x – 8 – 5 [Rearrange
terms, by exponent, with the constants last]
= 9x3 - 3x2 + 5x – 13
[Combine like terms]
Addition Practice
(½ x2 – 6x + ⅓) + (¾ x2 – 2x + ½)
=
SUBTRACTING Polynomials
Example:
(9x3 + 6x – 8) - (-3x2 – x – 5)
= 9x3 + 6x – 8 + 3x2 + x + 5 [Remove
parentheses. Change all the signs, because you
are subtracting]
Subtraction Practice
(x3 – 7x2y + 3xy2 - 2y3) - (2x3 + 4xy - 6y3)
=
Evaluate a Polynomial
Example: Evaluate 3x2 – 4x + 3 for x = -1
3x2 – 4x + 3
{GIVEN Problem}
= 3(-1)(-1) - 4(-1) + 3 {Substitute value for x}
= 3(1) + 4 + 3 {Apply PEMDAS rules, multiply}
= 3 + 4 + 3 {Complete multiplication}
= 10
{Addition, LEFT to RIGHT}
MULTIPLICATION of Polynomials
EXAMPLE: A binomial by a monomial
2a3(3a – 4) {Apply the distributive property}
=
MULTIPLICATION of Binomials
PROBLEM: (x + 3)(x + 5)
• FIRST: distribute the first term in the first
quantity (set of parentheses) to the second
entire quantity.
(x + 3)(x + 5)
x(x + 5) = x2 + 5x
MULTIPLICATION of Polynomials
PROBLEM: (x + 3)(x + 5)
• SECOND: distribute the second term (including
the sign) in the first quantity to the second entire
quantity.
(x + 3)(x + 5)
3(x + 5) = 3x + 15
MULTIPLICATION of Polynomials
PROBLEM: (x + 3)(x + 5)
• THIRD: combine like terms.
(x + 3)(x + 5)
= x2 + 5x + 3x + 15
= x2 + 8x + 15
PRACTICE
(x – 2)(x + 9)
=
FOIL Shortcut
What we have been doing has a nickname,
the FOIL method.
FOIL is an acronym for
First
Outer
Inner
Last
FOIL Example
GIVEN: (a + b)(c + d)
FIRST a*c
Outer a*d
Inner b*c
Last b*d
F O I
L
Example: (x+2)(x+3) = x2 + 3x + 2x + 6
= x2 + 5x + 6 [Simplify]
Special Binomial Products
• (x+a)(x-a) = x2 – a2
(difference of 2 squares)
• (x+a)2 = (x+a)(x+a) = x2 + 2ax + a2
(perfect square trinomial)
Special Products PRACTICE
(Sum and Difference of 2 numbers)
(x + 5)(x - 5)
=
(x + 9)(x - 9)
=
Special Products PRACTICE
(x+a)2 = (x+a)(x+a) = x2 + 2ax + a2
(perfect square trinomial)
EXAMPLE:
(x - 2)(x - 2)
=
Trinomial in the Product
(x – 3)(x2 – 2x + 6)
=
{USE distributive property}
PRACTICE
(x + 3)(4x2 + 6x – 5)
Trinomial by a Trinomial
(2x2 - 3x – 1)(4x2 + 6x – 5)
=
Multiply Several Binomials
(2x – 3)(x + 2)(x – 1)
=
PRACTICE