Lesson 7-6 Multiplying Polynomials by a Monomial
Example 1 Multiply a Polynomial by a Monomial
Find 3x(2x2 + 8x – 1).
Horizontal Method
3x(2x2 + 8x – 1)
= 3x(2x2) + 3x(8x) – 3x(1)
= 6x3 + 24x2 – 3x
Vertical Method
2x2 + 8x – 1
()
3x
6x3 + 24x2 – 3x
Original expression
Distributive Property
Multiply.
Distributive Property
Multiply.
Example 2 Simplify Expressions
Simplify 5b(2b – 4) + 4(b2 – 6b – 3).
5b(2b – 4) + 4(b2 – 6b – 3)
= 5b(2b) – 5b(4) + 4(b2) – 4(6b) – 4(3)
= 10b2 – 20b + 4b2 – 24b – 12
= (10b2 + 4b2) + [-20b + (-24b)] – 12
= 14b2 – 44b – 12
Original expression
Distributive Property
Multiply.
Commutative and Associative Properties
Combine like terms.
Standardize Test Example 3
Jenny pays a fee of $75 dollars to rent a car. She is charged an
additional $15 for each weekday and $20 for each weekend day she uses the car. Jenny rents
the car for 7 days, where w of those days are weekdays. What is the cost of Jenny’s car rental
if 5 of the 7 days were weekdays?
GRIDDED RESPONSE
Read the Test Item
You need to write an equation for Jenny’s car rental bill. Then evaluate to find the total cost.
Solve the Test Item
If w = the number of weekdays she rented the car, then 7 – w = the number of weekend days she
rented the car. Let C = the total charge for renting the car for the 7 days.
charge
=
fee
+
number of
weekdays
C
=
75
+
w
C = 75 + 15w + (7 – w)20
= 75 + 15w + 7(20) – w(20)
= 75 + 15w + 140 – 20w
= 215 – 5w


15
+
number of
weekend days
15
+
(7 - w)
Write the equation.
Distributive Property
Simplify.
Simplify.
An expression for Jenny’s car rental charge for
7 days is 215 – 5w, where w is the number of
weekdays she rents the car.
C = 215 – 5(5)
= 215 – 25
= 190
w=5
Multiply.
Subtract.
Jenny’s bill is $190. Grid in your response of 190.


1 90
C
20.
20
Example 4 Equations with Polynomials on Both Sides
Solve –2(2y + 7) –2y = 4(-y – 3) + 4.
-2(2y + 7) – 2y = 4(-y – 3) + 4
Original equation
-4y – 14 – 2y = -4y – 12 + 4
Distributive Property
-6y – 14 = -4y – 8
Combine like terms.
-14 = 2y – 8
Add 6y to each side.
-6 = 2y
Add 8 to each side.
-3 = y
Divide each side by 2.
The solution is –3.
Check
-2(2y + 7) – 2y = 4(-y – 3) + 4
Original equation
?
-2[2(-3) + 7] – 2(-3)  4[-(-3) – 3] + 4
y = -3
?
-2(-6 + 7) + 6  4(3 – 3) + 4
Simplify.
?
-2(1) + 6  4(0) + 4
Add and subtract.
?
-2 + 6  0 + 4
4=4 
Multiply.
Add.