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.
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.
|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
(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.
11 - w
(11 - w)w
The dimensions 5 x 6 or 6 x 5 has the greatest area.
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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