# AREA AND PERIMETER PROBLEMS INVOLVING POLYNOMIALS

Perimeter of the shape will be sum of measures of all sides around the figure.

Area of the shape can be figured out using the formula which related the given shape.

For example,

 Area of rectangleArea of squareArea of triangleArea of parallelogramArea of trapezoid Length x widthside x side(1/2) x base x heightbase x height(1/2) x h (a + b)

Problem 1 :

What is the distance around the rectangle if the length is

3x2 + 6x – 10

and the width is

3x + 5 ?

Solution :

The given shape is a rectangle, to find the distance around the shape rectangle is similar to find perimeter of the rectangle.

Length = 3x2 + 6x – 10

Width = 3x + 5

Perimeter of rectangle = 2(length + width)

= 2[(3x2 + 6x – 10) + (3x + 5)]

= 2[3x2 + 6x + 3x - 10 + 5]

= 2[3x2 + 9x - 5]

= 6x2 + 18x - 10

So, the length around the shape is 6x2 + 18x - 10.

Problem 2 :

If the perimeter of the pentagon below is 7x4 + 9x3 – 6x2 + 10, what is the length of the missing side?

Solution :

Number of sides of the shape pentagon = 5

Perimeter =  7x4 + 9x3 – 6x2 + 10

(3x4 - 4) + (5x3 - 2x2) + (4x3 + 6x) + (4x4 - 6x- 3x) + unknown side = 7x4 + 9x3 – 6x2 + 10

(3x4+ 4x4+ 5x3 + 4x3- 2x2 - 6x2 + 6x - 3x - 4) + unknown side = 7x4 + 9x3 – 6x2 + 10

(7x4+ 9x3 - 8x2 + 3x - 4) + unknown side = 7x4 + 9x3 – 6x210

unknown side

7x4 + 9x3 – 6x210 - (7x4+ 9x3 - 8x2 + 3x - 4)

7x4 - 7x4+ 9x- 9x3– 6x2 - 8x2 + 3x + 10 - 4

= -14x2 + 3x + 6

Problem 3 :

If the perimeter of the square below is 12x5 – 8x2 + 20x – 4, what is the length of one side?

Solution :

Perimeter of square = 12x5 – 8x2 + 20x – 4

Number of equal sides in the shape square = 4

Side length of square = Perimeter of square / 4

= (12x5 – 8x2 + 20x – 4) / 4

Factoring 4 from the numerator, we get

= 4(3x5 – 2x2 + 5x – 1) / 4

= 3x5 – 2x2 + 5x – 1

Problem 4 :

Ana knows that the perimeter of her backyard is (6x2 + 14x) feet. If the length of her backyard is (2x2 + 3x – 7) feet, what is the width of her backyard?

Solution :

Let w be the width of backyard. But it is not clear that what shape it is exactly.

Perimeter = 6x2 + 14x

Length = 2x2 + 3x – 7

2[(2x2 + 3x – 7) + w] = 6x2 + 14x

Dividing by 2 on both sides

[(2x2 + 3x – 7) + w] = (6x2 + 14x)/2

[(2x2 + 3x – 7) + w] = 3x2 + 7x

w = (3x2 + 7x) - (2x2 + 3x – 7)

w = 3x2 + 7x - 2x2 - 3x + 7

By combining like terms , we get

w = 3x- 2x2  + 7x - 3x + 7

w = x2+ 4x + 7

So, the required width of the rectangle is x2+ 4x + 7.

Problem 5 :

The area of the square below is represented by the expression 4x2 + 4x + 1. The area of the rectangle is represented by the expression x2 – 5x + 6. Using the diagram below, find the area of the shaded region.

Solution :

Area of the square = 4x2 + 4x + 1

Area of the rectangle = x2 – 5x + 6

= Area of the square - area of rectangle

= (4x2 + 4x + 1) - (x2 – 5x + 6)

= 4x2 - x2+ 4x + 5x + 1 - 6

= 3x2 + 9x - 5

Problem 6 :

A rectangular piece of wood has an area of

5x4 + 3x2 – 6x + 8

If two identical circles are cut out of the wood and the area of EACH circle is

x2 – 2

find the area of the remaining piece of wood. (Hint: Use the picture below.)

Solution :

Area of rectangular piece of wood = 5x4 + 3x2 – 6x + 8

Area of one circle = x2 – 2

Area of the remaining piece of wood

= Area of rectangle - 2(area of circle)

= 5x4 + 3x2 – 6x + 8 - 2(x2 – 2)

= 5x4 + 3x2 – 6x + 8 - 2x2 + 4

= 5x4 + 3x2 - 2x2– 6x + 8 + 4

= 5x4 + x2 – 6x + 12

So, area of the shaded part is 5x4 + x2 – 6x + 12.

Problem 7 :

A circular plot of land has an area of

7x5 – x3 + 4x2 + 9

If the walkway around this piece of land has an area of

x4 – 4x3 + 2x

what is the area of the land and walkway combined?

Solution :

Area of circular land = 7x5 – x3 + 4x2 + 9

Area of walkway = x4 – 4x3 + 2x

Combined area = (7x5 – x3 + 4x2 + 9) +(x4 – 4x3 + 2x)

= 7x5 + x4– x3 – 4x+ 4x+ 2x + 9

= 7x5 + x4– 5x3 + 4x+ 2x + 9

Problem 8 :

The width of Adrian’s bedroom is (x – 5) feet. He knows that the length is four times the width. Find the perimeter of Adrian’s bedroom.

Solution :

Width of the bedroom = x - 5

Length = 4(width)

= 4(x - 5)

Perimeter of the bedroom = 2(length + width)

= 4(x - 5) + (x - 5)

= 4x - 20 + x - 5

= (5x - 25) feet

So, the required perimeter is (5x - 25) feet.

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