 Hi, this is a video about polynomials to begin. We'll talk about monomials a monomial is the product of a Number and one or more variables raised to a whole number power So for instance, this is any number. Let's say three times a variable Let's use x raised to a power that is a whole number anything zero one two three four five So like three x to the eighth. That's an example of a monomial Three x to the negative second. That is not a monomial So to identify a monomial keep in mind that it cannot have any of the following It cannot have variables in the denominator or bottom of the fraction It cannot have a negative exponent on the variable and it cannot have fractions as exponents or a variable under a radical The coefficient of a monomial is the number in the front of the variable The degree of the monomial or term as it's sometimes referred to as is the exponent or power on the variable So let's identify whether each of the following is a monomial. If so, we will state its degree and its coefficient So in part a let's focus on what is the power on y it is three so part a is yes in this situation The degree is three that's the power on the variable and the coefficient is Negative seven that's the number out front in part B. Let's look at the power on the variable It's negative two since there is a negative power on the variable. This would be a no So we'll write no And that's because there is a negative exponent in part C. I have negative two square root of x And look at x. There's a radical surrounding it. So this is also no So no and that's because the variable is under a radical in part D I just have the number seven think about this one for a minute Okay, your minutes over so think about this more so as like seven Times the variable to the zero power that's seven times one which still has a value of seven Looking at it now you look at the power on the variable. It's zero, which is fine. That's a whole number. It's allowed So the answer is yes the degree is Zero and then the coefficient is We'll just say seven So why is monomial identification important? That's because that's what polynomials are made of a polynomial would be more than one Monomial or term connected with the plus or minus sign a Monomial is just one term 6x squared is an example a binomial is two terms 7x minus one is an example a trinomial is three terms 4x squared minus 6x plus two is an example The degree of a polynomial is the greatest degree of any term of the polynomial the leading coefficient Well, it's the coefficient of the term that determines the degree So we're gonna place a lot of emphasis on that term with the highest power When we say to write a polynomial in standard form It means to write it in descending order based on the power on the variable of each term So like 4x squared minus 6x plus two it's in standard form It's in descending order based on the power on the variable and the degree is to the leading coefficient is four So example two We'll identify each of the following by its number of terms then we'll identify the degree leading coefficient And we'll even write the polynomial in standard form So first start off in part a how many terms do we have well three three means we have a trinomial Now let's look at the term of highest power. That's negative 2t squared This means We have a degree of 2 and it means Still focusing on that terms coefficient. It means the leading coefficient is negative 2 Now let's talk about standard form. I have a second power First power no power that is standard form 2 1 nothing So it's already in standard form So already in standard form Looking at part B Still three terms Still a trinomial look at the term of highest power. That's positive 3x cubed The degree would be three The leading coefficient since it is the term of highest power is three With regard to standard form What is it well the 3x cubed has to come first Then the plus 7x There's a positive 7 and then the minus 1 in the end All right part C It has four terms. So it's not any it's not a monomial. It's not a binomial It's not a trinomial in general. We just say it is a polynomial The highest power term is the z to the seventh. So my degree would have to be seven Now the leading coefficient Remember when there's nothing in front of a variable. It's like saying there's a one there So my leading coefficient is one And then let's talk about standard form obviously the z to the seventh does come first but then the Plus six z to the fourth plus six z squared and then plus two So That is identifying polynomials and listing lots of information about them to end subtract polynomials will literally just combine like terms remember like terms would be the same variable raised to the same power our goal in this Subsection is to distribute drop parentheses combine like terms. So in part a There's nothing to distribute to either set of parentheses so I rewrite it Without parentheses and then we combine like terms. I have a minus 3x plus 7x which is 4x One minus two. It's negative one four x minus one is the answer In part B, there's nothing to distribute to the first set of parentheses, but the second set there is a minus or a minus one So in part B, we have 4x squared Minus 3x plus 3 Distribute the negative one Negative one times 2x squared is negative 2x squared Negative one times 7x is negative 7x negative one times negative one is plus one All right, do a little bit of cleanup here Combine like terms I have a 4x squared minus 2x squared Which is 2x squared. I have a minus 3x Minus 7x which is a minus 10x and 3 plus 1 plus 4 Last one part C. I've got two things to distribute here 2x times x is 2x to the second 2x times plus one It's plus 2x Negative three times x squared is negative 3x squared negative 3 times 4x is negative 12x Negative three times negative five is plus 15 combine like terms 2x squared minus 3x squared is negative x squared Plus 2x minus 12x is negative 10x and you follow through with the plus 15 at the end That's your answer for part C Well, I hope you enjoyed. Thanks for watching