1. Polynomial Multiplication
— Area and Volume
Topic 6.2.3

2. 6.2.3 1.1.1 Polynomial Multiplication — Area and Volume
Lesson
1.1.1
Topic
6.2.3
Polynomial Multiplication — Area and Volume
California Standards:
2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.
What it means for you:
You’ll multiply polynomials to solve problems involving area and volume.
Key words:
polynomial
monomial
distributive property

3. 1.1.1 6.2.3 Polynomial Multiplication — Area and Volume
Topic
6.2.3
Lesson
1.1.1
Polynomial Multiplication — Area and Volume
Polynomial multiplication isn’t just about abstract math problems.
Like everything in math, you can use it to work out problems dealing with everyday life.

4. 6.2.3 Polynomial Multiplication — Area and Volume
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Find Areas by Multiplying Polynomials
You can use polynomial multiplication to find the area of geometric shapes whose dimensions are expressed as polynomials.
For example, to find the area of this rectangle, you multiply its length by its width.
(2x + 11) cm
(x + 5) cm
Area = l × w = (2x + 11) × (x + 5)

5. 6.2.3 Polynomial Multiplication — Area and Volume
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Example 1
(5x + 6) inches
(3x + 2) inches x
Find the area of the space between the two rectangles:
Solution
The length of the middle rectangle is 5x + 6 – 2x = (3x + 6) in.
The width of the middle rectangle is 3x + 2 – 2x = (x + 2) in.
Area of space = area of large rectangle – area of small rectangle
= (5x + 6)(3x + 2) – (3x + 6)(x + 2)
= 15x2 + 10x + 18x + 12 – (3x2 + 6x + 6x + 12)
= 15x2 + 28x + 12 – 3x2 – 12x – 12
= (12x2 + 16x) in2
Solution follows…

6. 1.1.1 6.2.3 Polynomial Multiplication — Area and Volume
Topic
6.2.3
Lesson
1.1.1
Polynomial Multiplication — Area and Volume
Guided Practice
1. Find the area of a rectangle whose dimensions are (3x + 4) inches by (2x + 1) inches.
2x – 3
5x + 3
(3x + 4)(2x + 1) = 6x2 + 3x + 8x = (6x2 + 11x + 4) in2 1 2
2. Find the area of the triangle on the right.
(The formula for the area of a triangle is Area = bh.)
(0.5)(5x + 3)(2x – 3) = (0.5)(10x2 – 15x + 6x – 9) = (5x2 – 4.5x – 4.5) square units
3. The height of a triangle is (3x – 2) inches and its base is (4x + 10) inches. Find the area of the triangle.
(0.5)(4x + 10)(3x – 2) = (0.5)(12x2 – 8x + 30x – 20) = (6x2 + 11x – 10) in2
Solution follows…

7. 6.2.3 1.1.1 Polynomial Multiplication — Area and Volume
Lesson
1.1.1
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Guided Practice
4. Find the area of a rectangle whose dimensions are (3 + 2x) inches by (5 + 6x) inches.
5. Find the area of a square with side length (a2 + b2 – c2) ft.
6. The area of a trapezoid is given by A = ½h(b1 + b2) where h is the height and b1 and b2 are the lengths of the parallel sides. Find the area of this trapezoid.
(3 + 2x)(5 + 6x) = x + 10x + 12x2 = ( x + 12x2) in2
(a2 + b2 – c2)(a2 + b2 – c2) = a4 + a2b2 – a2c2 + a2b2 + b4 – b2c2 – a2c2 – b2c2 + c4 = (a4 + 2a2b2 – 2a2c2 + b4 – 2b2c2 + c4) feet2
(x + 1) in.
(x + 4) in.
x in.
(0.5)(x + 1)(x + x + 4) = (x + 1)(0.5)(2x + 4) = (x + 1)(x + 2) = (x2 + 3x + 2) in2
Solution follows…

8. 6.2.3 Polynomial Multiplication — Area and Volume
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Multiply Polynomials to Find Volumes
You’ve just used polynomials to find the areas of shapes.
You can also multiply polynomials to find volumes — the next two Examples show you how.

9. Multiply the first two polynomials Simplify the first product
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Example 2
Find the volume of the box on the right.
(5x + 8) in.
(6x – 4) in.
(4x + 6) in.
Solution
Volume = Length × Width × Height
= (5x + 8)(6x – 4)(4x + 6)
= [5x(6x – 4) + 8(6x – 4)](4x + 6)
Multiply the first two polynomials
= (30x2 – 20x + 48x – 32)(4x + 6)
= (30x2 + 28x – 32)(4x + 6)
Simplify the first product
Solution continues…
Solution follows…

10. 6.2.3 Polynomial Multiplication — Area and Volume
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Example 2
Find the volume of the box on the right.
(5x + 8) in.
(6x – 4) in.
(4x + 6) in.
Solution (continued)
= (30x2 + 28x – 32)(4x + 6)
= 4x(30x2 + 28x – 32) + 6(30x2 + 28x – 32)
Multiply out again
= 120x x2 – 128x + 180x x – 192
= 120x x x2 – 128x + 168x – 192
Commutative law
= (120x x2 + 40x – 192) in3

11. Multiply the first two polynomials
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Example 3
Find the volume of a box made from the sheet on the left by removing the four corners and folding.
6 in
8 in 2x
(6 – 4x) in
(8 – 4x) in
2x in
Solution
Volume = Length × Width × Height
= (8 – 4x)(6 – 4x)(2x)
= [8(6 – 4x) – 4x(6 – 4x)](2x)
Multiply the first two polynomials
= (48 – 32x – 24x + 16x2)(2x)
Solution continues…
Solution follows…

12. Simplify the first product Multiply by the third polynomial
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Example 3
Find the volume of a box made from the sheet on the left by removing the four corners and folding.
6 in
8 in 2x
(6 – 4x) in
(8 – x) in
2x in
Solution (continued)
= (48 – 32x – 24x + 16x2)(2x)
= (48 – 56x + 16x2)2x
Simplify the first product
= 96x – 112x2 + 32x3
Multiply by the third polynomial
= (32x3 – 112x2 + 96x) in3

13. 6.2.3 1.1.1 Polynomial Multiplication — Area and Volume
Lesson
1.1.1
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Guided Practice
7. Find the volume of a cube with side length (3x + 6) inches.
(3x + 6)(3x + 6)(3x + 6) = (9x2 + 36x + 36)(3x + 6) = (27x x x + 216) inches3
8. A concrete walkway around a swimming pool is 6 feet wide.
If the length of the pool is twice the width, x feet, what is the combined area of the walkway and pool?
x ft
6 ft
2x ft
(2x )(x ) = (2x + 12)(x + 12) = 2x2 + 24x + 12x = (2x2 + 36x + 144) ft2
Solution follows…

14. 1.1.1 6.2.3 Polynomial Multiplication — Area and Volume
Topic
6.2.3
Lesson
1.1.1
Polynomial Multiplication — Area and Volume
Guided Practice
Use the rectangular prism shown to answer these questions.
9. Find the volume of the prism.
10. Find the surface area of the prism.
11. If the height of the prism was reduced by 10%, what would be the new volume of the prism?
(2x + 3) ft
(3x – 1) ft
(x + 7) ft
(2x + 3)(3x – 1)(x + 7) = (6x2 + 7x – 3)(x + 7) = (6x3 + 49x2 + 46x – 21) ft3
2(2x + 3)(x + 7) + 2(3x – 1)(2x + 3) + 2(x + 7)(3x – 1) = (4x2 + 34x + 42) + (12x2 + 14x – 6) + (6x2 + 40x – 14) = (22x2 + 88x + 22) ft2
The volume of the new prism would be 90% of the volume of the old prism. 0.9 × (6x3 + 49x2 + 46x – 21) = (5.4x x x – 18.9) ft3
Solution follows…

15. 6.2.3 Polynomial Multiplication — Area and Volume Independent Practice
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Independent Practice
Expand and simplify the following.
1. (3y + 5)3
2. (2y – 1)3
3. The area of a parallelogram is given by the formula A = bh, where b is the length of the base and h is the height of the parallelogram.
Find the area of a parallelogram that has a base length of (2x2 + 3x – 1) cm and a height of (3x – 1) cm.
27y y y + 125
8y3 – 12y2 + 6y – 1
(6x3 + 7x2 – 6x + 1) cm2
Solution follows…

16. 6.2.3 Polynomial Multiplication — Area and Volume Independent Practice
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Independent Practice
(7x + 3) feet
(2x + 5) feet x
4. A gardener wants to put a walkway around her garden, as shown on the right.
What is the area of the walkway?
(14x2 + 16x) ft2
5. Obike made a box from a 10 inch by 9 inch piece of cardboard by cutting squares of x units from each of the four corners.
Find the volume of his box.
9 inches
10 inches x
(4x3 – 38x2 + 90x) in3
Solution follows…

17. 6.2.3 Polynomial Multiplication — Area and Volume Independent Practice
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Independent Practice
6. Find the volume of the solid shown.
(x + 4) ft
(x + 6) ft
( x3 + 7x2 + 32x + 48) ft3 1 2
7. Find the volume of another triangular prism that has the same base measurements as the one above but a height 25% less than the height shown above.
( x x2 + 24x + 36) ft3 3 8 4 21
Solution follows…

18. 6.2.3 Polynomial Multiplication — Area and Volume Round Up
Topic
6.2.3
Polynomial Multiplication — Area and Volume
Round Up
For problems involving area, you’ll have to multiply two terms or polynomials together.
For problems involving volume, you’ll have to multiply three terms or polynomials.