Download Sullivan College Algebra: Section R.5

Sullivan Algebra and
Trigonometry: Section R.4
Polynomials
Objectives of this Section
• Recognize Monomials
• Recognize Polynomials
• Add, Subtract, and Multiply Polynomials
• Know Formulas for Special Products
A monomial in one variable is the product
of a constant times a variable raised to a
nonnegative integer power. Thus, a
monomial is of the form:
ax k
where a is a constant, x is a variable, and k >
0 is an integer.
Examples of Monomials
Monomial
3x
4
2x
9
Coefficient
3
Degree
2
4
1
-9
0
A polynomial in one variable is an algebraic
expression of the form
an x  an 1 x
n
n 1
 a1 x  a0
where an , an1,, a1, a0 are constants, called
coefficients of the polynomial, n  0 is an
integer, and x is a variable. If an  0, it is
called the leading coefficient, and n is called
the degree of the polynomial.
Example:
Determine the coefficients and degree
of 2 x 4  3 x 2  x  5.
Coefficients: 2, 0, -3, 1, -5
Degree: 4
Polynomials are added and subtracted by
combining like terms.
Example: Addition
2 x
3
 
 x  8 x  1  3x  5x  2
2

 2 x  3x
3
3
 x
2
 5x  x  3x  1
3
2
3

 8 x  5 x   1  2 
Example: Subtraction
2 x
3
 
 x  8 x  1  3x  5x  2
2
3

 2 x  x  8 x  1  3x  5x  2
3

2
 2 x  3x
3
3
3
 x
2
 8 x  5 x   1  2 
  x 3  x 2  13 x  3
Polynomial multiplication can be done by using
the distributive property multiple times.
Example: Multiplication
 3 x  2  x  4 x  3
2
 3x  x  3x 4 x  3x  3  2  x  2 4 x  2  3
2
2
 3x 3  12 x 2  9 x  2 x 2  8x  6
 3x 3  14 x 2  17 x  6
Special Product Formulas
Difference of Two Squares
 x  a  x  a   x  a
2
2
Squares of Binomials, or Perfect Squares
2
2
x

a

x

2
ax

a


2
 x  a   x  2 ax  a
2
2
2
Special Product Formulas
Miscellaneous Trinomials
x  ax  b  x  a  bx  ab
2
ax  bcx  d  acx  ad  bcx  bd
2
Cubes of Binomials, or Perfect Cubes
 x  a   x  3ax  3a x  a
3
3
2
2
3
 x  a   x  3ax  3a x  a
3
3
2
2
3
Special Product Formulas
Difference of Two Cubes

2

2
x  a   x  a  x  ax  a
3
3
2

2

Sum of Two Cubes
x  a   x  a  x  ax  a
3
3
Polynomials in Two Variables
The degree of a polynomial in two variables
is the highest degree of all the monomials
with nonzero coefficients. The degree of
each monomial is the sum of the powers of
the variables.
Polynomial
2x 2 y 3
Degree
5
3x 2 - 4 xy 4
5
3a 3 - 2ab 2 + b 4
4