a rectangle has an area of ( x3+x2-10x+8) square meters, and a width of (x+4) meters. Find its length.
You probably meant to say
x^3+x^2-10x+8
since area = l x w
x^3+x^2-10x+8 = length(x+4)
length = (x^3+x^2-10x+8)/(x+4)
= x^2 - 3x + 2
(I used synthetic division, you could also use long algebraic division)
To find the length of the rectangle, we need to divide its area by its width. In this case, the area of the rectangle is given as x^3 + x^2 - 10x + 8 and the width is given as x + 4.
Using the formula for the area of a rectangle, we can write the equation:
Area = Length * Width
Substituting the given values, we have:
x^3 + x^2 - 10x + 8 = Length * (x + 4)
To find the length, we need to isolate it on one side of the equation. Let's do that step by step:
x^3 + x^2 - 10x + 8 = Length * (x + 4)
x^3 + x^2 - 10x + 8 = Length * x + Length * 4
x^3 + x^2 - 10x + 8 = Length * x + 4Length
Now, let's isolate Length by moving the terms involving Length to one side:
x^3 + x^2 - 10x + 8 - Length * x - 4Length = 0
Grouping like terms:
(x^3 + x^2 - x) - (10x + 4Length - 8) = 0
Factoring out x:
x(x^2 + x - 1) - (10x + 4Length - 8) = 0
At this point, we can see that it won't be easy to directly factor out Length from the expression. At this point, we can use polynomial long division to find the quotient and remainder of the polynomial x(x^2 + x - 1) divided by (x + 4):
x(x^2 + x - 1)
-----------------------
x + 4 | x^3 + x^2 - x + 4x + 8
- (x^3 + 4x^2)
------------
-3x^2 - x
+ ( -3x^2 - 12x )
----------------
11x + 8
Now we have:
(x + 4) * (x^2 + 4x - 3) - (11x + 8) = 0
Expanding the first part:
x^3 + 4x^2 - 3x + 4x^2 + 16x - 12 - 11x - 8 = 0
Combining like terms:
x^3 + 8x^2 + 2x - 20 = 11x + 8
Rearranging the terms:
x^3 + 8x^2 + 2x - 11x - 20 - 8 = 0
Simplifying:
x^3 + 8x^2 - 9x - 28 = 0
The equation above is a cubic equation. To solve it, we can use numerical methods such as factoring, synthetic division, or iterative methods like the Newton-Raphson method. Unfortunately, finding an exact solution for this equation is not straightforward.
However, if you are looking for an approximate solution, you can use software or online calculators to find the roots of the equation or plot the graph of the equation to estimate the value of x. Once you have the value of x, you can substitute it back into the expression (x + 4) to find the length of the rectangle.