The area of a desktop in the shape of a rectangle is given by the function f(x)=2x2+5x+3

Question

. The width of the desktop is given by the function g(x)=x+1
. What function represents the length of the desktop?

Enter the missing values.

(1 point)
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Remaining Attempts : 3

Answer 1

To determine the length of the desktop, we can use the relationship that the area $A$ of a rectangle is equal to the product of its length $L$ and width $W$.

In this case, we are given:

• The area function $f(x)=2x^2+5x+3$
• The width function $g(x)=x+1$

The area can be expressed as: $$A=L\times W$$ where $L$ is the length and $W$ is the width.

We can rearrange this equation to solve for the length $L$: $$L=\frac{A}{W}=\frac{f(x)}{g(x)}$$

Substituting the given functions: $$L=\frac{f(x)}{g(x)}=\frac{2x^2+5x+3}{x+1}$$

Now we need to simplify this expression. We can perform polynomial long division or factor if possible.

1. Factor $2x^2+5x+3$: We look for two numbers that multiply to $2\times 3=6$ and add to $5$. These numbers are $2$ and $3$. We can rewrite: $$2x^2+2x+3x+3=2x(x+1)+3(x+1)=(2x+3)(x+1)$$

2. Thus, we have: $$2x^2+5x+3=(2x+3)(x+1)$$

3. Now substituting back into our expression for length: $$L=\frac{(2x+3)(x+1)}{x+1}$$

4. Cancel out $x+1$ (assuming $x+1\ne 0$): $$L=2x+3$$

Therefore, the function that represents the length of the desktop is: $$L(x)=2x+3$$

Now, you can fill in the blank:

The function that represents the length of the desktop is: $$L(x)=\text{Fill this with }2x+3$$