# Multiplication of Polynomials Worksheet

Multiplication of Polynomials Worksheet
• Page 1
1.
The length and width of a rectangular board are in the ratio 3 : 2. The frame adds 7 inches to the width and 8 inches to the length to form a laminated board. Write a polynomial expression that represents the total area of the board, including the frame.
 a. 6$y$2 - 21$y$ - 56 b. 6$y$2 + 37$y$ + 56 c. 4$y$2 + 37$y$ - 6 d. $y$2 + $y$ + 3

#### Solution:

If length of the board is 3y, then the width of the board is 2y
[Lenght : width = 3 : 2.]

After lamination, the length of the rectangular board is 3y + 8 and its width is 2y + 7.
[Length is increased by 8 and width by 7.]

Let A be the area of the laminated rectangular board.

A = (length) × (width)
[Area of a rectangle.]

A = (3y + 8)(2y + 7)
[Replace the variables with the values, given.]

A = 3y(2y + 7) + 8(2y + 7)
[Distribute (2y + 7) to each 3y and 8.]

A = 6y2 + 21y + 16y + 56
[Group the like terms.]

A = 6y2 + 37y + 56
[Combine like terms.]

2.
Evaluate: ($a$ + $z$) ($a$ - $z$) ($a$2 + $z$2)
 a. - $a$4 - $z$4 b. $a$2 - $z$2 c. $a$4 + $z$4 d. $a$4 - $z$4

#### Solution:

(a + z) (a - z) (a2 + z2)

= (a2 - z2) (a2 + z2)
[Using special product: (a + z) (a - z) = a2 - z2.]

= (a2)2 -(z2)2
[Using special product: (a + z) (a - z) = a2 - z2.]

= a4 - z4
[Simplify.]

3.
Find:
(2$x$$a$ + 6$y$$b$) (3$x$$a$ + 5$y$$b$)
 a. 6${x}^{2a}+28{x}^{a}{y}^{b}+30{y}^{2b}$ b. 6${x}^{2a}+28{x}^{a}{y}^{b}+31{y}^{2b}$ c. 5 d. 6${x}^{2a}-28{x}^{a}{y}^{b}-30{y}^{2b}$

#### Solution:

(2xa + 6yb)(3xa + 5yb)

= 2xa (3xa + 5yb) + 6yb (3xa + 5yb)
[Distributive property.]

= 6x2a + 10xayb + 18xayb + 30y2b
[am · an = am + n exponent property.]

= 6x2a + 28xayb + 30y2b
[Simplify.]

4.
Find the area of the figure.

 a. (2$y$2 + 4$y$ +1) sq units b. (4$y$2 + 4$y$ +1) sq units c. (2$y$2 + 2$y$ +1) sq units d. (4$y$2 - 4$y$ +1) sq units

#### Solution:

The two adjacent sides of the figure are equal.

So, the figure is a square with side 2y + 1.

Area of a square = side2
[Write the formula.]

= (2y + 1)(2y + 1)
Area of the square shown in the figure = (2y + 1)2
[Substitute the value of the side and express the square as product of terms.]

= 2y(2y + 1) + 1(2y + 1)
[Use distributive property.]

= 4y2 + 4y + 1
[Multiply the coefficients and add the exponents.]

Area of the figure is (4y2 + 4y +1) sq units.

5.
The height of the triangle is 8 times the base $b$ in a right triangle. Find the area of the triangle.
 a. 4$b$2 square units b. 4$b$ square units c. 12$b$2 square units d. 8$b$2 square units

#### Solution:

Height of the triangle is 8b units.

Area of a triangle = 1 / 2 x base x height
[Write the formula.]

= 12 x b x 8b
[Substitute the values.]

= 4b2
[Simplify.]

Area of the right triangle is 4b2 square units.

6.
What is the degree of the polynomial obtained by calculating the area of a square having a side (3$y$ + 3)?
 a. 1 b. 4 c. 2 d. 3

#### Solution:

Area of the square = side x side
[Write the formula.]

= (3y + 3) x (3y + 3)
[Substitute the value.]

= 3y(3y + 3) + 3(3y + 3)
[Use distributive property.]

= 9y2 + 9y + 9y + 9
[Multiply.]

= 9y2 + 18y + 9
[Combine like terms.]

Highest exponent of y in the obtained polynomial is 2.

The degree of the polynomial obtained by calculating the area of the square is 2.

7.
Find the area of the unshaded region in the figure.

 a. 23 square units b. 32 square units c. 45 square units d. 56 square units

#### Solution:

Side of the larger square is 6 units.

Side of the shaded portion is 2 units.

Area of the square with side 6 = (6 x 6) square units

Area of the shaded portion = (2 x 2) square units

Area of the unshaded portion = Area of larger square - Area of the shaded portion

= (6 x 6) - (2 x 2)
[Substitute the values.]

= 36 - 4 = 32
[Multiply and subtract.]

Area of the unshaded portion = 32 square units

8.
Find the area of the rectangle.

 a. (4$y$2 - $y$) square units b. (4$y$2 - 4$y$) square units c. ($y$2 - 4) square units d. ($y$2 - 4$y$) square units

#### Solution:

2y and (2y - 2) are the sides of the rectangle.

Area of the rectangle = Length x Width
[Write the formula.]

= 2y(2y - 2)
[Substitute the values.]

= 2y(2y) - 2(2y)
[Use distributive property.]

= 4y2 - 4y
[Multiply.]

Area of the rectangle is (4y2 - 4y) square units.

9.
Find the area of the rectangle, if the width of the rectangle is 4 more than $\frac{1}{4}$ its length.
 a. ($L$2 + 4$L$) square units b. ($L$2/4 + $L$) square units c. ($L$2/4 - 4$L$) square units d. ($L$2/4 + 4$L$) square units

#### Solution:

Let L be the length of the rectangle and ( L / 4 + 4 ) be the width of the rectangle.

Area of the rectangle = Length x Width
[Write the formula.]

= L ( L4 + 4)
[Substitute the values.]

= L ( L4 ) + L (4)
[Use distributive property.]

= L2/4 + 4L
[Multiply.]

Area of the rectangle is (L2/4 + 4L) square units.

10.
Write (5$y$2 + 20$y$) as a product of two factors.
 a. 5$y$($y$ - 4) b. 5$y$2($y$ + 4) c. 5$y$($y$ + 4) d. 5($y$2 - 5)

#### Solution:

5y2 + 20y
[Original polynomial.]

5y2 = 5 x y x y
[Express as factors.]

20y = 4 x 5 x y
[Express as factors.]

5y2 = 5y x y, 20y = 4 x 5y
[Arrange the terms.]

5y2 + 20y = 5y(y + 4)
[Use distributive property.]