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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.
6y2 - 21y - 56
b.
6y2 + 37y + 56
c.
4y2 + 37y - 6
d.
y2 + 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.]


Correct answer : (2)
 2.  
Evaluate: (a + z) (a - z) (a2 + z2)
a.
- a4 - z4
b.
a2 - z2
c.
a4 + z4
d.
a4 - z4


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.]


Correct answer : (4)
 3.  
Find:
(2xa + 6yb) (3xa + 5yb)
a.
6x2a+28xayb+30y2b
b.
6x2a+28xayb+31y2b
c.
5x2a+26xayb- 30y2b
d.
6x2a-28xayb-30y2b


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.]


Correct answer : (1)
 4.  
Find the area of the figure.

a.
(2y2 + 4y +1) sq units
b.
(4y2 + 4y +1) sq units
c.
(2y2 + 2y +1) sq units
d.
(4y2 - 4y +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.


Correct answer : (2)
 5.  
The height of the triangle is 8 times the base b in a right triangle. Find the area of the triangle.
a.
4b2 square units
b.
4b square units
c.
12b2 square units
d.
8b2 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.


Correct answer : (1)
 6.  
What is the degree of the polynomial obtained by calculating the area of a square having a side (3y + 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.


Correct answer : (3)
 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


Correct answer : (2)
 8.  
Find the area of the rectangle.


a.
(4y2 - y) square units
b.
(4y2 - 4y) square units
c.
(y2 - 4) square units
d.
(y2 - 4y) 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.


Correct answer : (2)
 9.  
Find the area of the rectangle, if the width of the rectangle is 4 more than 1 4 its length.
a.
(L2 + 4L) square units
b.
(L2/4 + L) square units
c.
(L2/4 - 4L) square units
d.
(L2/4 + 4L) 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.


Correct answer : (4)
 10.  
Write (5y2 + 20y) as a product of two factors.
a.
5y(y - 4)
b.
5y2(y + 4)
c.
5y(y + 4)
d.
5(y2 - 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.]


Correct answer : (3)

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