Transcript Prealgebra & Introductory Algebra
3.3
Introduction to Polynomials
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Defining Term and Coefficient
Term
– a number or a product of a number and variables raised to powers
Coefficient
– numerical factor of a term
Constant
– term which is only a number
Polynomial
is a finite sum of terms of the form
ax
n , where
a
is a real number and
n
is a whole number.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
Defining Term and Coefficient
In the polynomial 7
x
5 +
x
2
y
2 – 4
xy
+ 7 There are 4
terms
: 7
x
5 ,
x
2
y
2 , – 4
xy
and 7.
The
coefficient
of term 7
x
5 is 7, of term
x
2
y
2 is 1, of term –4
xy
is –4 and of term 7 is 7.
7 is a
constant
term.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
Types of Polynomials Monomial
is a polynomial with exactly one term.
Binomial
is a polynomial with exactly two terms.
Trinomial
is a polynomial with exactly three terms.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
Degrees
The
degree of a term in one variable
is the exponent on the variable.
The
degree of a constant
is 0.
The
d egree of a polynomial
is the greatest degree of any term of the polynomial. The degree of 9
x
3 – 4
x
2 + 7 is 3.
The
degree of a term with more than one variable
is the sum of the exponents on the variables. The degree of the term 5
a
4
b
3
c
be written as
c
1 ).
is 8 (remember that
c
can Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
Evaluating Polynomials
Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved.
Example
Find the value of 2
x
3 – 3
x
+ 4 when
x
= – 2.
2
x
3 – 3
x
+ 4 = 2( – 2 ) 3 – 3( – 2 ) + 4 = 2( – 8) + 6 + 4 = – 6 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
Combining Like Terms
Like terms
are terms that contain exactly the same variables raised to exactly the same powers.
Warning!
Only like terms can be combined through addition and subtraction.
Example
Combine like terms to simplify.
x
2
y = x
2
y
+
xy
– + 10
x
2
y y
+ 10 +
xy x
2
y
– 2
y
+
xy
–
y
+
xy
– 2
y
= (1 + 10) = 11
x
2
y x
2
y
+ 2
xy
+ (1 + 1)
xy
– 3
y
+ (– 1 – 2)
y
Group like terms.
Use the distributive property.
Simplify.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed