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### Course: Algebra 1 (Eureka Math/EngageNY)>Unit 4

Lesson 1: Topic A: Lessons 1-2: Factoring monomials

# Intro to factors & divisibility

Sal explains what it means for a polynomial to be a factor of another polynomial, and what it means for a polynomial to be divisible by another polynomial.

## Want to join the conversation?

• But how did he get 10x in the first place? In the first problem, he only multiplied the similar terms (3 times -2, x times x etc.) But in the second one he multiplied everything
• When multiplying binomials, think of it as doing the distributive property. Multiply each term by each term. So x * x = x^2, while 3 * 7 = 21. But, x * 7 =7x, while 3 * x = 3x. So, x^2 +7x + 3x + 21. Simplifying that, you add the 3x to the 7x to equal 10x. The final answer is, x^2 + 10x + 21
• Hi,
Just to get some clarity,
What is the difference between binomial, polynomial, trinomial, etc?
• A monomial is a polynomial with 1 term.
A binomial is a polynomial with 2 terms.
A trinomial is a polynomial with 3 terms.
• can decimals be factors or is it just integers? as an example, you can definitely say that 3 is a factor of 6, but can you say that 2.5 is a factor of 5?
• Great question! When we talk about factors of whole numbers, we are looking for whole numbers. 2.5 is not a whole number, so it is not a factor of 5. Negative integers, like -1 and -5, are not factors either, because they are not whole numbers. The only factors of 5 are 1 and 5, making 5 prime.
• What exactly is the purpose of factorizing something?

Great Question! Similarly in algebra, factoring is a remarkably powerful tool, which is used at every level. It provides a standard method for solving quadratic equations as well, of course, as for simplifying complicated expressions. It is also useful when graphing functions. Factoring (or factorising) is the opposite of expanding.

This is a great process of simplification. Also, factoring is a complementary operation to the distributive property, it is a way to “unpack” the multiplication done by applying the distributive property. Reorganizing polynomials by factoring allows us to find solutions for certain types of polynomials.

Hope this helps.
• Why did he put 10x at ??
(1 vote)
• When you multiply (x+3)(x+7), you would get x squared plus 7x plus 3x plus 21. You have 3x and 7x, which add together to get 10x.
• If x^2 + 10x + 21 is a trinomial, then is x^2 + 3x + 7x + 21 a quadrinomial, or still a trinomial, since it can be simplified to the former?
• In its expanded form, it is a quadrinomial because it has 4 terms at that point. However, it is still a quadratic expression.
• Is not (3xy)(-2x^2y^3) the same as [(xy) + (xy) + (xy)][(-x^2y^3) + (-x^2y^3)] ?
• Yes, but why do you want / need to write the expression that way? This is rarely done.
• Would 1/2 and 20 be factors of 10? Could fractions and decimals be defined as factors? It doesn't specify if the factors must be whole numbers or integers. Also, if you have 3xy for example, and you say that 6xy^2 is a factor of it, would that be wrong? If so, is it because a monomial cannot have a fractional coefficient or is it because factors cannot be decimals or fractions?
(1 vote)
• Factors are integers that will divide evenly into a number without leaving a remainder. 1/2 would not be considered a factor of 10 as it is not an integer. 20 would not be a factor of 10 because it does not divide evenly into 10.
• Does anyone know if, in an expression such as 2(x-5)(x+3), would "2" be considered a factor or just a coefficient?
• You can think of it as either. Calling it a factor suggests that you're examining the product of those three terms, and calling it a coefficient suggests that you're looking the the constant factor of other, similar expressions as well.
• what is the difference between polynomials, and binomials?
• A polynomial is simply a fancy word to describe a math statement with several terms. A mononomial is the opposite - it is a statement with only one term. A binomial is a type of polynomial that contains exactly two terms. All of these are pretty simple to remember because they use the same Greek/Latin roots that many other math terms use. (Poly means many, mono means one, and bi means two). Here are some examples of all of these concepts.

Polynomial:
- 4x^2 + 5x + 6
- x + a + c
- 433 + 19b + 17c + 25a

Mononomial:
- 5x
- a
- 19
- 7

Binomial:
- x + 7
- 4x + 9
- 9a + b

Hope that helps!