Multiply the expressions in parts (a) through (c) to remove the parentheses.

1. $(x − 1)(x + 1)$ 

   Hint (a):

   Use a generic rectangle to multiply.

   Show rectangle (a):

   In a rectangle, $\left(\text{length}\right)\left(\text{width}\right) = \text{area}$. Find the area of each small rectangle inside the generic rectangle. 2 by 2 square, left edge labeled, top, x, bottom, minus 1. Bottom edge labeled, left, x, right, + 1.

   Show step (a):

   Added labels to interior: top left, x squared, top right, 1, x, bottom left, 1, x, bottom right, negative 1.

   Answer (a):

   The total area of the generic rectangle, or $x^2−1$. 

   

   

2. $2x(x + 1)(x + 1)$ 

   Hint (b):

   First, multiply $2x$ and $(x+1)$.Then use a generic rectangle to multiply that result by $(x+1)$.

   Show rectangle (b):

   2 by 2 square, left edge labeled, top, 2, x squared, bottom, + 2, x, Bottom edge labeled, left, x, right, + 1.

   Show areas (b):

   Added labels to interior: top left, 2, x to the third, top right, 2, x squared, bottom left, 2, x squared, bottom right, 2, x.

   Answer (b):

   $2x^3+4x^2+2x$ 

   

   

3. $(x − 1)(x + 1)(x − 2)$ 

   Hint (c):

   $(x^2−1)(x−2)$
   Now use a generic rectangle.

4. Find the $x$- and $y$-intercepts of $y = x^3 − 2x^2 − x + 2$. The factors in part (c) should be useful.

   Hint (d):

   The $x$-intercepts are where $y=0$.

   Step 1(d):

   $0=x^3−2x^2−x+2$

   More Help (d):

   The $y$-intercept is where $x=0$:

   $y=0^3−2(0)^2−0+2$

   Step 2(d):

   $0=(x−1)(x+1)(x−2)$ from part (c).

   Step 3(d):

   Using the Zero Product Property: $x−1=0\ \ \text{or}\ \ x+1=0\ \ \text{or}\ \ x−2=0$

   Answer (d):

   $x=1\ \ \text{or}\ \ x=−1\ \ \text{or}\ \ x=2$