Consider the rectangle with an area of 24x^3-54x^2-15x, with a side length of 6x^2-15x. Write and simplify an expression for the length of the rectangle.
^ Could someone please show me how to do this?
It's just an exercise in polynomial long division. In fact, looking more closely, it's even easier.
area = length * width
length = area/width
Now, 24x^3-54x^2-15x
= x(24x^2-54x-15)
= 3x(8x^2 - 18x - 5)
= 3x(2x-5)(4x+1)
6x^2 - 15x
= 3x(2x-5)
So, length = 3x(2x-5)(4x+1) / 3x(2x-5)
= 4x+1
To find the length of the rectangle, we need to divide the area by the width. In this case, the width of the rectangle is given as 6x^2 - 15x.
To find the length, we divide the area (24x^3 - 54x^2 - 15x) by the width (6x^2 - 15x):
Length = Area / Width
Length = (24x^3 - 54x^2 - 15x) / (6x^2 - 15x)
Now, let's simplify this expression using polynomial long division. Here are the steps to perform long division:
1. Start by dividing the highest degree term of the numerator (24x^3) by the highest degree term of the denominator (6x^2).
This gives us 4x (since 24x^3 ÷ 6x^2 = 4x).
2. Multiply the divisor (6x^2 - 15x) by the quotient you just found (4x). This gives us 24x^3 - 60x^2.
3. Subtract the result obtained in step 2 from the numerator (24x^3 - 54x^2 - 15x) to get the remainder: (-54x^2 + 15x).
4. Bring down the next term from the numerator, which is -54x^2.
5. Repeat steps 1-4 using the new numerator (-54x^2 + 15x) and the same denominator (6x^2 - 15x).
Following these steps, we get:
Length = 4x - 9
Therefore, the simplified expression for the length of the rectangle is 4x - 9.