Let's use the following example to learn this new part: x^2 + 6x + 5.
First, we need to make a tile image of the shape below. Go ahead and do that! Now that you have that, we want to figure out how to write out the length and width.
We know that the dimensions of the big square (x^2) is x by x.
We know that the dimension of the rod (x) is x by 1.
We know that the dimension of the little square (1) is 1 by 1.
Let's look at the length of the image below (the left side of the shape). We are only looking at the side lengths of the shape, not the whole area. Looking at the left side, I see 2 green rods and 1 blue square. The dimension of one rod is x by 1. The left side of the shape has the smaller part of the rod, 1. There are two of them, so we have 1 and 1 to represent the green parts of the length. The blue parts of the length is part of the square. The dimension of one large square is x by x. The one blue part that is part of the length is one dimension: x.
So, in total for the length, we have 1 x and 2 1s. I can write that more easily as x and 2 = x + 2.
We can apply the same thinking to find the width. We see one long blue side length (x) and 3 small green parts (1s). In total for the width, we have 1 x and 3 1s. I can write that more easily as x and 3 = x + 3.
So, my length is (x + 2) and my width is (x + 3).
To practice with the dimensions, draw rectangles with each shape below AND indicate the length and width.
1. x^2 + 7x + 6
2. x^2 + 4x + 4
3. 2x^2 + 8x + 6
4. 2x^2 + 7x + 3
5. 2x^2 + 8x + 8
6. 3x^2 + 10x + 3
7. 3x^2 + 8x + 5
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