The length of a rectangle is (x^2-2x-1) feet and the width is (2x + 3) feet. Express the area as a polynomial.

Question

The length of a rectangle is (x^2-2x-1) feet and the width is (2x + 3) feet. Express the area as a polynomial.

A right triangle has one leg that is x feet long and another leg that is x+2 feet long. The hypotenuse is 2x-7 feet long. What are the sides of the triangle? (Round to 2 decimal places.)

Answer 1

A = L * w
= (x^2-2x-1) * (2x + 3)
= (2x)(x^2) + (2x)(-2x) + (2x)(-1) + (3)(x^2) + (3)(-2x) + (3)(-1)
= 2x^3 - 4x^2 - 2x + 3x^2 - 6x - 3
= 2x^3 - x^2 - 8x - 3 feet^2



L1 = x
L2 = x+2
H = 2x-7

Pythagorean Theorem:
(L1)^2 + (L2)^2 = H^2
(x)^2 + (x+2)^2 = (2x-7)^2
x^2 + x^2 + 4x + 4 = 4x^2 - 28x + 49
4x^2 - 28x + 49 - x^2 - x^2 - 4x - 4 = 0
2x^2 - 32x + 45 = 0
Quadratic formula:
x = [32 +/- sqrt((-32)^2 - 4*2*45)] / 4
x = [32 +/- sqrt(1024 - 360)] / 4
x = [32 +/- sqrt(664)] / 4
x = [32 +/- sqrt(4*166)] / 4
x = [32 +/- 2*sqrt(166)] / 4
x = [16 +/- sqrt(166)] / 2
Use a calculator to get decimal form.

Answer 2

The area of a rectangle is given by multiplying the length times the width. In this case, the length is x^2 - 2x - 1 and the width is 2x + 3. So, multiply these numbers:

Area = (2x + 3)(x^2 - 2x - 1)

First, you will multiply 2x to each term in the second parentheses, then you will multiply 3 to each term in the second parentheses:

Area = (2x)(x^2 - 2x - 1) + (3)(x^2 - 2x - 1)
Area = (2x^3 - 4x^2 - 2x) + (3x^2 - 6x - 3)

You can drop the parentheses and collect like terms:

Area = 2x^3 - x^2 - 8x - 3

And that is your area expressed as a polynomial.





For the triangle, use the Pythagorean theorem: a^2 + b^2 = c^2

(x)^2 + (x+2)^2 = (2x-7)^2

A number squared is just a number times itself, so you can rewrite this to look like:

x^2 + (x+2)(x+2) = (2x-7)(2x-7)

Now, use foil to multiply it out:

x^2 + (x^2 + 2x + 2x + 4) = (4x^2 -14x - 14x + 49)

Now you can drop the parentheses and collect like terms:

2x^2 + 4x + 4 = 4x^2 - 28x + 49

Subtract (4x^2 - 28x + 49) from both sides to set this equal to 0:

2x^2 + 4x + 4 - 4x^2 + 28x - 49 = 0
-2x^2 + 32x - 45 = 0

Use the quadratic formula to solve for x:

x = [-32 +- sqrt(32^2 - 4(-2)(-45))]/2(-2)
x = [-32 +- sqrt(1024 - 360)]/(-4)
x = [-32 +- 25.768]/(-4)
x = 14.442 or x = 1.558

Now, x = 1.558 cannot be correct since the hypotenuse would be a negative number (length can't be negative), so the answer is x = 14.442

The lengths of your sides would be 14.442, 16.442, and the hypotenuse would be 21.884

Answer 3

1. Area of rectangle = Length * Width

A=(x^2-2x-1) * (2x+3)
A=2x^3-4x^2-2x+3x^2-6x-3
A=2x^3-x^2-8x-3
This is a polynomial equation of 3 degree.

2. For a right angled triangle
Hypotenuse^2 = A^2 + B^2
Where A=One leg of triangle & B=Another leg of triangle.
Therefore,
(2x-7)^2 = (x)^2 + (x+2)^2
4x^2 - 28x + 49 = x^2 + x^2 + 4x + 4
2x^2 - 32x + 45 = 0
Following the eqaution :
x=(-b + sqrt(b^2-4ac))/2a & x=(-b-sqrt(b^2-4ac))/2a

where a=2, b=-32, c=45
Therefore,
x=(32+sqrt(1024-360))/4 & x=(32-sqrt(1024-360))/4
=> x=8+6.44=14.44 & x=8-6.44=2.44

Answer 4

Question 1
A(x) = (2x + 3) (x² - 2x - 1)
A(x) = 2x³ - 4x² - 2x + 3x² - 6x - 3
A(x) = 2x³ - x² - 8x - 3

Question 2
(2x - 7)² = x² + (x + 2)²
4x² - 28x + 49 = x² + x² + 4x + 4
4x² - 28x + 49 = 2x² + 4x + 4
2x² - 32x + 45 = 0
x = [ 32 ±√(1024 - 360) ] / 4
x = [32 ±√(664)] / 4
x = [ 32 ± 25.77 ] / 4
x = 14.44 , x = 1.56
Accept x = 14.44 to give positive value for length of sides.
x = 14.44 ft
x + 2 = 16.44 ft
2x - 7 = 21.89 ft

Answer 5

note
right triangle

a^2 + b^2 = c^2
(x + 2)^2 + x^2 = (2x - 7)^2
x^2 + 4x + 4 + x^2 = 4x^2 - 28x + 49
2x^2 + 4x + 4 = 4x^2 - 28x + 49
2x^2 - 32x + 45 = 0

quadratic formula

x = [-b +/- sqrt(b^2 - 4ac)] / (2a)

x = [32 +/- sqrt(664)] / 4
x = [32 +/- 2sqrt(166)] / 4
x = [16 + sqrt(166)] /2 = [16 + 12.88] /2 = 14.44 feet, or
x = [16 - sqrt(166)] /2 = [16 - 12.88] /2 = 1.55 feet