Exponents and Polynomials - Clinton Community College · PDF file4x2 This IS a polynomial...
Transcript of Exponents and Polynomials - Clinton Community College · PDF file4x2 This IS a polynomial...
Learning Objectives 4-4
Polynomials
Monomials, binomials, and trinomials
Degree of a polynomials
Evaluating polynomials functions
Polynomials
Polynomials are sums of these "variables and exponents"
expressions.
Each piece of the polynomial, each part that is being
added, is called a "term". Polynomial terms have
variables which are raised to whole-number exponents (or
else the terms are just plain numbers).
There are no square roots of variables, no fractional
powers, and no variables in the denominator of any
fractions.
Polynomials
6x –2 This is NOT
a polynomial term...
...because the variable has a
negative exponent.
This is NOT
a polynomial term...
...because the variable is in the
denominator.
This is NOT
a polynomial term...
...because the variable is inside a
radical.
4x2 This IS a polynomial term... ...because it obeys all the rules.
y
x2
x
Polynomial Degrees
Second-degree polynomial, 4x2, x2 – 9, or
ax2 + bx + c
Third-degree polynomial, –6x3 or x3 – 27
Fourth-degree polynomial, x4 or 2x4 – 3x2 + 9
Fifth-degree polynomial, 2x5 or x5 – 4x3 – x + 7
Monomial
An expression containing only one term is called a
monomial.
Example: 7, x, 7x, -6x, ab, etc.
A monomial is a number, a variable, or the product of
a number and one or more variables with whole
number exponents.
Not a monomial: 8 + x, 2/n, 5x, a-1, -3x-3, x2.5
Binomial and Trinomial
Binomial: An expression containing two terms is
called a binomial.
Examples: 7x+5, 6y - p
Trinomial: An expression containing three terms is
called a trinomial.
Examples: 2x+3y-4z
Monomial, Binomial, and Trinomial
Type Definition Example
Monomial A polynomial with one term 5x
Binomial A polynomial with two terms 5x - 10
Trinomial A polynomial with three terms
Evaluating Polynomial Functions
“Evaluating” a polynomial is the same as evaluating
anything else: you plug in the given value of x, and
figure out what y is supposed to be.
Evaluate f(x) = 2x3 – x2 – 4x + 2 at f(-3)
Evaluating Polynomial Functions
The revenue ($) that a mfg. of desks receives is given
by the polynomial function: f(d) = -0.08d2 + 100d
where d is the number of desks.
a) Find the total revenue if 625 desks are made.
b) Does increasing the number of desks being made to 650
increase the revenue?
Section 4.4 Review
Polynomials
Monomials, binomials, and trinomials
Degree of a polynomials
Evaluating polynomials functions
Section 4.5 Learning Objective
Adding and subtracting monomials
Adding and subtracting polynomials
Adding and subtracting multiples of polynomials
An application of adding polynomials
Remember Like Terms?
4x and 3 NOT like terms The second term has no variable
4x and 3y NOT like terms
The second term now has a variable,
but it doesn't match the variable of
the first term
4x and 3x2 NOT like termsThe second term now has the same variable,
but the degree is different
4x and 3x LIKE TERMSNow the variables match and the
degrees match
Adding and Subtracting Monomials
Step 1: Remove the ( ).
Step 2: Combine like terms.
Examples:
4ab + (-2ab) =
4ab - (-2ab) =
6x2 - x2 =
Application #1
A house is purchased for $105,000 and is expected to
appreciate $900 per year, its value y after x years is
given by the polynomial function
f(x) = 900x + 105,000.
a) What is the expected values in 10 years?
Application #2
A house second home is purchased for $120,000 and
is expected to appreciate $1,000 per year.
a) Find a polynomial function that will give the
appreciated value y of the house in x years.
b) Find the value of this second house after 12 years.
Section 4.5 Review
Adding and subtracting monomials
Adding and subtracting polynomials
Adding and subtracting multiples of polynomials
An application of adding polynomials
Section 4.6 Learning Objectives
Multiplying monomials
Multiplying a polynomial by a monomial
Multiplying a binomial by a binomial
The FOIL method
Special products
Multiplying a polynomial by a binomial
Multiplying three polynomials
Multiplying binomials to solve equations
Multiplying Monomials
When multiplying two monomials, multiply the
numerical factors and then multiply the variable
factors.
Example:
(5x2y3)(6x3y4)30x33y7
Multiplying a Polynomial by a Monomial
Use the distributive property to remove parentheses
and simplify.
Example:
2x3(3x2 – 5x)
Multiplying a Binomial by a Binomial
Multiply each term of one binomial by each term of
the other binomial and combine like terms.
Example:
(x + 3)(x + 2)
(x + 3y)(2x − 5y)
The FOIL Method
F First terms
O Outside terms
I Inside terms
L Last terms
One way to keep track of your distributive property is to Use the FOIL
method. Note that this method only works on (Binomial)(Binomial).
Find the product of (z + 3)(z + 1)
Special Products
Square of the sums:
(x + y)2 = X2 + 2xy +y2
The square of the differences:
(x – y )2 = X2 – 2xy +y2
Product of the sum and difference of two terms:
(a + b)(a – b) = a2 – b2
Multiplying a polynomial by a binomial
Rule: To multiply one polynomial by another, multiply
each term of one polynomial by each term of the
other polynomial and combine like terms.