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  
5y^{2}  y^{2}  5x + 2x^{2}  
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.
x^{2} + 5x^{2}  3x^{2} = 1 + 5  3 = 3x^{2}
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 onemeter 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 onehalf 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)
= 6x^{2}  10x
6. Simplifying Polynomials
3x(5  12x)  x(10 + 4x)
= 15x  16x^{2}  10x  4x^{2
}
= 5x  20x^{2}
x(4x  15) + 8x(2x  7)
= 4x^{2} + 15x + 16x^{2}  56x
= 12x^{2}  41x