Number Systems, Polynomials, and Algebra

The Number System

• Honestly, if you are in Algebra 2, you should know this by now, but here’s a brief overview.

• Natural numbers are numbers like 1, 3, 5, etc. There are no negative numbers, decimals, or the number 0.

• Whole numbers are basically all natural numbers, but you include 0.

• Integers are numbers that don’t have fractions (includes all negative numbers that are whole, 0, and all positive numbers that are whole.)

• Irrational numbers are numbers that have a fraction, no matter where they are on the number line.

Dividing Polynomials

How to divide polynomials with long division:

The idea of this is that you want to separate the parts of the problem down. So for example, we want to see what would multiply to x to make 2x^2. As a result, we get 2x. We then multiply it to the divisor, and subtract. We also put the 2x up onto the quotient. We do the same for the next part.

How to divide polynomials with synthetic division:

To do this, we first want to change our divisor to see what x equals to. X + 2 = 0, so x = -2. Now we put -2 as our divisor. Then, we take the coefficients of each number from the dividend, and put it on our table. We bring the first coefficient down, then multiply by -2. We get -4, so we subtract 4 from 7. We get 3, so we multiply that by -2. We get -6, so we subtract 6 from 6. This is 0. Now when fixing our terms back to standard form, we just take out x. If there's a remainder left however, set it over the divisor (ex 2x + 3 + 3/x+5).

The Remainder Theorem

• When f(x) is divided (x-a), the remainder is f(a).

  

The Factor Theorem

• If f(a) = 0, then (x - a) is a factor.