This weeks unit, Quadratics: Multiplying & Factoring, was once again a great refresher unit. Though I don’t feel like I understand quadratics and polynomials 100%, I now have a much better understanding of what they are and how to expand and/or factor them.

As far as I know and as I’ve mentioned in a previous post, a polynomial is an umbrella term that includes monomials, binomials, trinomials and quadratics. From what KA has told me, polynomial essentially means “many” (poly-) “terms” (nomial). That is somewhat misleading however since a monomial is one term (as you might guess, a binomial contains two terms and a trinomial contains three terms). A polynomial is the sum of a finite number of terms where each term has a coefficient being multiplied by a variable and being raised to non-negative, integer power. Ex:

• Ax^n
  • A = the coefficient
  • x = the variable
  • n = the exponent
  • Ex. 3x^2
• Monomial
  • 4x^2
  • 4
    • 4 could be considered a monomial since you could add 1 as variable x and 1 as exponent n (4[1^1])
• Binomial
  • 4x + 2
  • 4x^2 + 3x
• Trinomial
  • x^2 – 10x + 25

I’m still not 100% certain what a quadratic is but I believe it is a second degree term (i.e. there’s a term to the power of two and no term with a higher power) that has two other terms. The Trinomial example above would therefore be considered a quadratic.

I learned about three things that are not considered polynomials:

1. Polynomials cannot have a negative exponent (ex.  10x^-7 + 9),
2. They cannot have a fraction/decimal exponent (ex. 9a^1/2), and
3. They cannot have a variable as an exponent (ex. 9a^a).

The phrase “to the Nth degree” refers to the highest exponent in a polynomial. The term that’s raised to the highest power dictates the degree the polynomial is referred to as. For example:

• 10x^7 = 7^th degree polynomial
• 4x^3 = 3^rd degree polynomial
• 2y^2 = 2^nd degree polynomial
• 99g^99 = 99^th degree polynomial

Standard form when writing polynomials is to write them with the highest degree first and going lower in degree with each successive term.  For example, 4 + 8x^3 – 7x^2 would be written as 8x^3 – 7x^2 + 4. Another important phrase when it comes to polynomials is the phrase Leading Coefficient which is the first number in a polynomial. In the example 8x^3 – 7x^2 + 4, the Leading coefficient would be 8.

The unit then got into multiplying binomials and factoring trinomials. I began by working on multiplying binomials using the distributive property:

• (x + 2)(x + 3)
• = x(x + 3) + 2(x + 2)
• = x^2 + 3x + 2x + 4
• = x^2 + 5x + 4

I then moved on to factoring polynomials, particularly focusing on factoring trinomials. I was given a “Factoring Checklist” to go through when factoring polynomials which essentially is a step by step process you go through to factor properly.

1. Is there a greatest common factor?
   1. 6x^2 + 3x = 3x(2x + 1)
2. Is there a difference of squares?
   1. x^2 – 16 = (x – 4)(x + 4)
   1. 25x^2 – 9 = (5x + 3)(5x – 3)
3. Is the polynomial of the form x^2 + bx + c? If so, use the sum product pattern:
   1. x^2 + 7x + 12
   1. = x^2 + 3x + 4x +12
   1. = (x^2 + 3x) + (4x + 12)
   1. = x(x + 3) + 4(x + 3)
   1. = (x + 4)(x + 3)
4. If the polynomial is of the form ax^2 + bx + c, use the grouping method:
   1. 4x^2 + 16x + 15
      1. a * c = 4 * 15 = 60
      1. b = 16
      1. 60 = 6 * 10
      1. 16 = 6 + 10
   1. = 4x^2 + 6x + 10x + 15
   1. = (4x^2 + 6x) + (10x + 15)
   1. = 2x(2x + 3) + 5(2x + 3)
   1. (2x + 5)(2x + 3)

There’s also a method of factoring trinomials called the Perfect Square method which is used when both a and c in a polynomial that takes the form ax^2 + bx + c are perfect squares. I’m still not 100% sure on how the method works but this would be my best description of how it works at this point:

• ax^2 + bx + c
  • b = 2 *square root of*ac
• (ax + c)^2
• 25x^2 -30x + 9
• = (5x)^2 -2(5)(3)x + 3^2
• =(5x – 3)^2

I think the above example may be wrong but at this point I really don’t understand this method and don’t know how to work through it. My hope is I’ll be given these types of questions later on and the concept will start to sink in then.

The unit I’m moving on to in Week 16 is Quadratics Functions & Equations (460/2700 M.P.). I went through the first few videos/questions which showed me examples of trinomials being graphed as parabolas. I’m assuming that’s what the gist of the unit will be about which I think will really help me to better understand the theory/the point behind trinomials, quadratics etc. I certainly hope so anyways since, even though I understand polynomials much better after finishing Week 15, I still feel like I don’t understand the point of them… (*angry emoji face*)