Question

The area (in square inches) of a rectangle is given by the polynomial function A(b) = b2 + 6b+ 8. If the width of the rectangle is (b + 2) inches, what is the length?

Answer 1

The area of a rectangle is the product of its length and its width.

Since the area of the rectangle is given by th polynomial function:

`A(b)=b^2+6b+8`

And the width of the rectangle is given by the binomial:

`(b+2)`

Factor out (b+2) from the expression of the area to find the other binomial factor, which corresponds to the length.

To do so, notice that the coefficient of b^2 is 1 and the constant term is 8. The product of those two numbers, 1 and 8, is 8.

The factors of 8 that sum to 6 are 2 and 4. Then, write the term 6b as 2b+4b:

`A(b)=b^2+2b+4b+8`

Factor b from th first two terms and 4 from the last two terms:

`\begin{gathered} b^2+2b+4b+8=b(b+2)+4(b+2) \\ =(b+4)(b+2) \end{gathered}`

If we identify (b+4) as the length of the rectangle we can see that the area is given by the product of (b+4) and (b+2), which are the length and the width of the rectangle.

Therefore, the length of the rectangle is:

`b+4`