This document provides examples of multiplying algebraic expressions, including: 1) Multiplying monomials and polynomials 2) Using tables, vertical, horizontal, and FOIL methods 3) Solving a multi-step problem involving designing a skate park and finding its area given dimensions

2. Multiplication with Algebra Tiles
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3. Example 1 Multiply monomials
Find the product 3x( – 2x3 ).
3x ( – 2x3 ) = 3 • ( –2 ) • x • x3 Regroup factors.
= – 6x4 Simplify.

4. Example 2 Multiply a monomial and a polynomial
Find the product 2x3 ( x3 + 3x2 – 2x + 5 ).
2x3 ( x3 + 3x2 – 2x + 5) Write product.
= 2x3 ( x3 ) + 2x3 ( 3x2 ) – 2x3 ( 2x ) + 2x3 ( 5 ) Distributive
property
= 2x6 + 6x5 – 4x4 + 10x3 Product of
powers property

5. Example 3 Multiply polynomials using a table
Find the product ( x – 4 ) ( 3x + 2 ) .
SOLUTION
STEP 1 Write subtraction as addition:
( x – 4 ) ( 3x + 2 ) = [ x + ( – 4)] ( 3x + 2 )
STEP 2 Make a table of products.
3x 2 3x 2
x 3x2 x 3x2 2x
–4 –4 – 12x –8

6. Example 3 Multiply polynomials using a table
ANSWER
The product is 3x2 + 2x – 12x – 8, or 3x2 – 10x – 8.

7. Example 4 Multiply polynomials vertically
Find the product ( b2 + 6b – 7 ) ( 3b – 4 ).
SOLUTION
STEP 1 Multiply by – 4. STEP 2 Multiply by 3b.
b2 + 6b – 7 b2 + 6b – 7
× 3b – 4 × 3b – 4
– 4b2 – 24b + 28 – 4b2 – 24b + 28
3b3 + 18b2 – 21b

8. Example 4 Multiply polynomials vertically
STEP 3 Add products.
b2 + 6b – 7
× 3b – 4
– 4b2 – 24b + 28
3b3 + 18b2 – 21b
3b3 + 14b2 – 45b + 28

9. Example 5 Multiply polynomials horizontally
Find the product ( 2x2 + 5x – 1 ) ( 4x – 3 ).
( 2x2 + 5x – 1 ) ( 4x – 3 ) Write product.
= 2x2 ( 4x – 3 ) + 5x ( 4x – 3 ) – 1 ( 4x – 3 ) Distributive
property
= 8x3 – 6x2 + 20x2 – 15x – 4x + 3 Distributive
property
= 8x3 + 14x2 – 19x + 3 Combine like
terms.

10. FOIL

11. Example 6 Multiply binomials using the FOIL pattern
Find the product ( 3a + 4 ) ( a – 2 ).
( 3a + 4 ) ( a – 2 )
= ( 3a ) ( a ) + ( 3a ) ( – 2 ) + ( 4 ) ( a ) + ( 4 ) ( – 2 ) Write products
of terms.
= 3a2 + ( – 6a ) + 4a + ( – 8 ) Multiply.
= 3a2 – 2a – 8 Combine like
terms.

12. Example 7 Solve a multi-step problem
SKATEBOARDING
You are designing a rectangular skateboard park
on a lot that is on the corner of a city block. The park
will be bordered by a walkway along two sides. The
dimensions of the lot and the walkway are shown in the
diagram.
a. Write a polynomial that represents the area of the
skateboard park.
b. What is the area of the park if the walkway is 3 feet
wide?

13. Example 7 Solve a multi-step problem
SOLUTION
a. Write a polynomial using the formula for the area of
a rectangle. The park’s length is the lot’s length
minus the width of the sidewalk, or 45 – x . Similarly,
the park’s width is 33 – x .

14. Example 7 Solve a multi-step problem
Area = length • width Formula for area of a
rectangle
= ( 45 – x ) ( 33 – x ) Substitute for length
and width.
= 1485 – 45x – 33x + x2 Multiply binomials.
= 1485 – 78x + x2 Combine like terms.

15. Example 7 Solve a multi-step problem
CHECK You can use a graph to check
the solution. Use a graphing
calculator to display the
graphs of y1 = ( 45 – x ) ( 33 – x )
and y2 = 1485 – 78x + x2 in the
same viewing window. Because
the graphs coincide, you know
that the product of 45 – x and 33 – x
is 1485 – 78x + x2 .
b. Substitute 3 for x and evaluate.
Area = 1485 – 78 ( 3 ) + ( 3 ) 2 = 1260
ANSWER The area of the park is 1260 square feet.

16. 9.2 Warm-Up (Day 1)
Find the product.
1. 3
5x (3x) 2. x(7x + 4)
2
Find the product using a table.
3. (4n −1)(n + 5)

17. 9.2 Warm-Up (Day 2)
Find the product using the vertical method.
1. (x 2 + 2x +1)(x + 2)
Find the product using the horizontal method.
2. (3y − y + 5)(2y − 3)
2
Find the product using the FOIL method.
3. (4b − 5)(b − 2)