Download Section 10

Section 10.1 – Addition and Subtraction of Polynomials
Recall that we defined a term as a number, variable, product of numbers
and/or variables, or a quotient of numbers and/or variables.
A term is called a monomial if there is no division by a variable.
ie, 3x2, 7a,
xy2, 19
A monomial or a sum or difference of monomials is called a polynomial.
ie, 5x2 + 3x – 1,
ab2 – ½a, y + 19
Adding Polynomials: Combine like terms
Examples: Add the following polynomials
a) (9x – 3) + (2x + 6)
b) (4x2 + 9x – 10) + (–2x3 + 4x + 1)
c) (ab2 + 9ab – 13b) + (15b + 2ab2 – 10)
Opposites of Polynomials: Two polynomials are opposites (additive
inverses) if their sum is zero. We can find the opposite of a polynomial by
replacing each term with its opposite, (ie, change the sign of every term).
Examples: Find the opposite polynomial
a) 3x2 + 7x – 9
b) –x – y
Subtracting Polynomials: Subtracting means “adding the opposite”
Examples: Subtract the following polynomials
a) (6x + 1) – (–7x + 2)
b) (8t2 – 5t + 7) – (3t2 – 2t + 1)
c) (0.5y4 – 0.6y2 + 0.7) – (2.3y4 + 1.8y – 3.9)
Evaluating Polynomials and Applications
Examples: Evaluate each of the following when x = 2 and x = –3.
a) 7 – x + 3x2
b) –3x3 + 7x2 – 3x – 2
Example: In a sports league of n teams in which all teams play each other
twice, the total number of games played is given by the polynomial n2 – n.
How many total games would be played if a league has 10 teams?
Section 10.2 – Multiplying and Factoring Polynomials
See pg. 711 for a visual representation of multiplying polynomials
The Product Rule for Exponents:
For any number a and any positive integers m and n, a m  a n  a mn
Examples: Multiply
a) (3x)(2x) =
b) (5x)( –3y) =
c) a4 ∙ a6 =
d) (3a2)(6a9) =
e) (–4x2y3)(3x6y7) =
When a polynomial contains two terms, it is called a binomial.
Examples: Multiply
a) 2x (3x + 2) =
b) 5x (2x2 – 3x + 8) =
c) –5x2y (3x3y5 – 7xy5) =
Factoring is the reverse of multiplying. To factor an expression means to
find an equivalent expression that is a product.
ab + ac = a (b + c)
Example: Factor the following
a) 6x + 5x =
b) 8b – 12 =
c) 3x + 12y – 3 =
d) –4x + 8y – 16z =
e) 18a + 12b – 24 =
f) 15x6 + 25x4
g) 10ab3 + 5a2b – 15ab
Section 10.3 – More Multiplication of Polynomials
Multiplying Two Binomials:
Example: Multiply using the distributive property:
(x + 4) (x + 3) =
Example: Multiply using the distributive property:
(2x + 3) (x – 5) =
We can use the word FOIL to help us remember the products when
multiplying two binomials:
F = first
O = outer
I = inner
L = last
Examples: Multiply using FOIL
a) (x + 6) (2x – 7) =
b) (m + 5) (2m – 9) =
A polynomial containing three terms is called a trinomial.
Example: Multiply
(x + 4) (x2 + 2x – 3) =
Example: Multiply
(3x – 1) (4x2 – 2x – 1) =
Section 10.4 – Integers as Exponents
Exponent Rules:
b1 = b
for any number b.
b0 = 1
for any non-zero number b.
a n  1n for any non-zero numbers a and b and any integer n.
a
a
 
b
n
 
  b 
a
 
n
for any non-zero numbers a and b and any integer n.
Examples: Simplify
a) 170 =
b) (–98.6)1 =
Examples: Write an equivalent expression using only positive exponents.
Then simplify.
a) 2–4 =
b) (–3)–5 =
c) x–7 =
d) –8y–2 =
e) (5a)–3 =
f)
 3
 
7
 
2

Section 10.5 – Scientific Notation
Scientific Notation for a number is an expression of the form M × 10n
where n is an integer and 1 ≤ M < 10.
Examples: Convert each number to scientific notation
a) 485,000,000
b) 0.00065
c) 14,670,000
Examples: Convert each number to decimal notation
a) 7.853 × 10–5
b) 7.853 × 105
c) 10–7
Example: The total revenue of NASCAR is expected to be $3423 million by
2006. Convert this number to scientific notation.