Week 3 Day 2 (Continue Algebra including Dividing Polynomials)

Operations with Polynomials

With polynomials, addition and subtraction are still combinations of like with like. You may want to put the different terms in the same order (e.g., ascending order) and then stack the two polynomials so that the like terms are in the same columns. In the case that one polynomial has a term raised to a power not represented in the other polynomial, you can use a placeholder with a coefficient of zero.

When multiplying polynomials, you must multiply each term in one polynomial by each term in the other polynomial and then simplify if necessary.

The acronym “FOIL” or “First, Outer, Inner, Last” may be useful here (and it makes a smiley face).

Example:       

First: 3x * 2x = 6x²

Outer: 3x * 2a = - 6xa

Inner: a * 2x = 2xa

Last: a * 2a = - 2a²

All together: 6x² – 6xa + 2xa – 2a²

Simplified: 6x² – 4xa – 2a²

 

When dividing polynomials by monomials, we just divide each term in the polynomial by the monomial until all terms in the polynomial have been divided, and simplify where possible. Sometimes quick cancelations are possible, too.

 

When dividing polynomials by other polynomials, it is better to arrange the dividend and divisor in descending order before doing long division. If a power is missing from the dividend, use 0 as a coefficient and include the variable with that power. Step by step, the process is like this:

1. Divide the first term in the dividend by the first term in the divisor.

2. Multiply the result by both terms in the divisor.

 

3. Subtract the product from the dividend.

 

4. Repeat until each term in the dividend has been divided.

 

5. Any remainder should be represented as a fraction where the numerator is the remainder and the denominator is the divisor.

 

Using zeros as coefficients to represent missing powers looks like this for the following problem:

(m³ – m) ÷ (m + 1) =

 

Remember: Wednesday next week is the midterm, review on Monday!