Algebra Review - Polynomials

1. Finding Maximum Areas of Rectangles with Tables

To find the maximum area of a rectangle with tables, you must keep plugging in numbers to test it. For example, if the total perimeter is 16, what is the greatest area possible? First make a table:

Length (l) Width (w) Area (l w)

 

The dimensions 4 x 4 has the greatest area.

1 7 7
2 6 12
3 5 15
4 4 16
5 3 15
6 2 12
7 1 7

2. Combining Like Terms

An expression is something like: 5x + 2x + x
Terms are: 5x, 2x, and x
Like terms have the same variable. Here 5x, 2x, and x are like terms because they all have x.

Like Terms

 

Unlike Terms

x + 2x + 8x   5y + 2xy + x
5y2 - y2   5x + 2x2
5xy + .5xy - 9xy   x + y + z

When you combine terms such as x + 2x + x + 2x, there is always a 1 in front of an x. This is called the coefficient. The coefficient of 2x is 2. Normally we do not write 1 in front of x. So to add x + 2x + x + 2x, it is just 1 + 2 + 1 + 2 = 6 = 6x.

x2 + 5x2 - 3x2 = 1 + 5 - 3 = 3x2


3. Adding and Subtracting Polynomials

Polynomials are terms, such as x or y, or expressions with more than one term, such as x + y.

2x + 3y + 4x + 5y = 2x + 4x + 3y + 5y
                         = 6x + 8y

5a - (-4a) = 5a + 4a
               = 9a

(3x + 4y - 8z + 16) - (2x - 5y - 8)
                                = 3x + 4y - 8z + 16 - 2x + 5y + 8
                                = x + 9y - 8z + 24


4. Maximizing Areas

Example: You have 22 one-meter sections of fencing to enclose a garden. The dimensions must be in whole numbers. What is the greatest area possible?

Expression: l + w = 11 (because the length and width is one-half the perimeter)

A = l w
   = (11 - w)w    (substitute by solving l + w = 11 for l)

Then use the table and plug in values to find the maximum area.

Width
w
Length
11 - w
Area
(11 - w)w

 

The dimensions 5 x 6 or 6 x 5 has the greatest area.

1 10 10
2 9 28
3 8 24
4 7 28
5 6 30
6 5 30
7 4 28
8 3 24
9 2 18
10 1 10

5. Multiplying Out Polynomials

If you have something like a(b + c), you can turn it to ab + ac by multiplying it out.

3(4a + b) = 3(4a) + 3b
              = 12a + 3b

2x(3x - 5) = 2x(3x) + 2x(-5)
               = 6x2 - 10x


6. Simplifying Polynomials

3x(5 - 12x) - x(10 + 4x)
                = 15x - 16x2 - 10x - 4x2
                = 5x - 20x2

-x(4x - 15) + 8x(2x - 7)
                = -4x2 + 15x + 16x2 - 56x
                = 12x2 - 41x

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