CHAPTER 5-CONTINUED
a) 14  9 x 2  9 x  3x 5
I) SECTION 5.3: POLYNOMIALS;
ADDITION AND SUBTRACTION OF
POLYNOMIALS
OBJECTIVES: You will learn
1)Identify polynomials and understand the
vocabulary used to describe them.
2) Evaluate polynomials
3) Add and subtract polynomials
4) Understand polynomials in several
variables
1) Identify polynomials and understand
the vocabulary used to describe them.
a) Polynomial: a polynomial in a single
variable x is the sum of terms of the form
ax n , where a is a real number and n is a
whole number
Ex:
b)Degree of a term: the degree of each term
in a polynomial in a single variable is equal
to the variable’s exponent. The degree of a
constant term is _____
Monomial:
Binomial:
Trinomial:
Ex1: List the coefficient of each term in the
given polynomial
 6 x 5  12 x 3  9 x 2  9 x  5
b) 2 x 5  x 8  23  4 x 3  x  7 x 2
2) Evaluate polynomials
Ex3: Evaluate the polynomial for the given
value of the variable
 3x 4  8 x 2  12 x for x = -2
Ex4: Evaluate the given polynomial function
f ( x)  2 x 3  5 x 2  10 x  25 , f(2)
3) Adding and Subtracting Polynomials:
Ex5: Add or subtract
a)
(4 x 5  7 x 3  3x 2 )  (8x 5  5x 3  2 x 2  7)
b)
(2 x 9  5x 4  7 x 2 )  (6 x 6  4 x 5  23x 2 )
c) Degree of a polynomial
The degree of the leading term is also called
the degree of the polynomial
Ex2: Rewrite the polynomial in descending
order. Identify the leading term, the leading
coefficient, and the degree of the polynomial
Ex6: For the given functions f(x) and g(x),
find f(x)+g(x), and f(x)-g(x)
1
f ( x)  x 3  8 x  31, g ( x)  4 x 3  x 2  3x  25
b) The average cost per shirt, in dollars, to
produce x T-shirts is given by the function
f ( x)  0.00017 x 2  0.06 x  13.124 . What is
the average cost per shirt to produce 100 Tshirts?
4) Polynomials in Several Variables
Ex7: Evaluate the polynomial
a) 3x 2  11xy  y 2 for x=4, and y =3
II)SECTION 5.4: MULTIPLYING
POLYNOMIALS
b) 7 x 2 yz  9 xy 2 z  10 xyz 2 for x=2, y=3,
z= -1
OBJECTIVES: You will learn
1)Multiply monomials
2)Multiply a monomial by a polynomial
3)Multiply polynomials
4)Find special products
1)Multiplying Monomials
i)Degree of each term in the polynomial:
is equal to sum of the exponents of its
variable factors.
Ex8: For each polynomial, list the degree of
each term and the degree of the polynomial.
 4 x 8 y 6  3x 3 y 7  x 5 y 11  6 x 2 y 5 z 4
Ex1: Multiply
a) 8 x 2  3x 3  10 x 5
b)  5 x 4 y 3 z  14 x 2 y 9 z 3
Ex2: Find the missing monomial
2 x 9  ____  24 x16
Ex9: Add or subtract
2)Multiply a monomial by a polynomial:
a)
4
2 5
3 2
4
2 5
3 2 multiply the monomial by each term of the
( x y  7 x y  15 y z t )  (8x y  11x y  9 y z t )
polynomial
Ex3: Multiply
a)  4(4 x  9)
2
b) 2 x 2 y 4 (3x 4  6 xy  7 y 5 )
Ex4: For the given functions f(x) and g(x),
find f ( x)  g ( x)
f ( x)  x 4  7 x 2  8 x  5 ; g ( x)  5x 4
3)Multiply polynomials
FOIL method: (a+b)(c+d)
4)Find special products
(a  b)( a  b) 
Ex4: Multiply:
a) (3x  2)( x  8)
(a  b) 2 
(a  b) 2 
b) (7 x  5)(2 x 2  3x  15)
Ex5: Find the special product using the
appropriate formula
a) (2 x  5)( 2 x  5)
c) ( x 2  3x  2)(3x 2  x  5)
b) (2 x  5) 2
c) ( x  5) 2
d) ( x  7 y )( 2 x  3 y )
Ex6: Find the missing factor or term
a) ( x  5)  ________  x 2  25
b) (_______) 2  x 2  12 x  36
c) (________) 2  9 x 2  12 x  4
3
III)SECTION 5.5: DIVIDING
POLYNOMALS
OBJECTIVES: You will learn
1)Divide a monomial by a monomial
2)Divide a polynomial by a monomial
3)Divide a polynomial by a polynomial by
using long division
4)Use placeholders when dividing a
polynomial by a polynomial
Ex3: Divide. Assume all variables are
nonzero.
22 x 5  36 x 4  14 x 2
a)
2x 2
b)
2a 10 b 8  8a 5 b 9  10a 6 b
2a 5 b
1)Divide a monomial by a monomial
Ex1: Divide. Assume all variables are
nonzero
 24 x 9 y 10
a)
6x 5 y 8
b) (72 x y )  (10 x y )
13
8
8
6
Ex2: Find the missing monomial. Assume
x0
20 x 10
 5 x 4
a)
?
b)
24 x10
4x 7

?
3
2) Divide Polynomials by Monomials
To divide a polynomial by a monomial,
divide each term of the polynomial by the
monomial
Ex4: Find the missing dividend or divisor.
Assume x  0
10 x 8  50 x 5  35 x 3
 2 x 5  10 x 2  7
a)
?
b)
?
 x 2  9x  1
5
7x
3) Divide a polynomial by a polynomial
by using long division
Step1: Divide the term in the dividend with
the highest degree by the term in the divisor
with the highest degree. Add this result to
the quotient
Step2: Multiply the monomial obtained in
step 1 by the divisor, writing the result
underneath the dividend.(Align like terms
vertically)
Step3: Subtract the product obtained in step
2 from the dividend.
Step4: Repeat steps 1-3 with the result of
step 3 as the new dividend. Keep repeating
this procedure until the degree of the new
dividend is less than the degree of the
divisor
Ex5: Divide using long division
4
a) ( x 2  15x  56)  ( x  7)
b)
b)
x 3  27
x3
x 3  11x 2  37 x  17
x 1
c) ( x 5  3x 3  5 x 2  x  3)  ( x  2)
4)Use placeholders when dividing a
polynomial by a polynomial
Ex6: Divide using long division
x 3  12 x  11
a)
x3
5