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The parts of polynomial expressions

Discover the magic of polynomials! Learn to identify terms, coefficients, and exponents in a polynomial. Understand that terms are the parts being added, coefficients are the numbers multiplying the powers of x, and exponents are the powers to which x is raised. Dive into the world of polynomials and make math fun! Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user Tony Bible
    Could someone explain to me what concept a polynomial represents? Why is it special? What can i do with it that I can't do with some other concept? What are the limits of what can be done with a polynomial, i.e. the constraints?

    Until I know what a polynomial actually is I can't move on to learning about its constituent parts. I can't find this information anywhere. It is almost as if nobody actually knows what a polynomial really is. Jumping in and telling me about its constituent parts or simply defining it with regards to these constituent parts is not what I'm looking for.

    A referential analogy would be great. For instance: a sine wave is nothing more and nothing less than a means of describing fluid oscillating motion as it appears in nature. Light, sound and electricity are all connected by the fluid oscillating nature of a sine wave. Since the sine wave is a fluid and non-linear type of motion it is a necessary component in the creation of and understanding of curved geometric shapes in the real or modeled world, i.e one complete oscillation of the sine wave is a sufficient component to help describe and/or create two-dimensional curved geometric shapes and two complete oscillations are a necessary component to help describe and/or create 3 dimensional curved geometric shapes.

    I understand that the above analogy may not be perfect. My mathematical skills are poorly developed at this point. I will give you that BUT it does help me to understand a fundamental principal and move forward having received an answer.

    Kind regards and thanks,

    (98 votes)
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    • leaf blue style avatar for user Stefen
      Wow! What a fantastic question!

      Polynomials have been around for a long time, but the name polynomial has only been in use since around the 17th century. Polynomials evolved directly from word problems. Way, way back, farmers, economists and kings (that is to say, anyone involved in business) used to describe and solve their problems with words. So back then, what we would now call a polynomial equation would be written out using words, for example, “2 plots of carrots, 3 plots of peas and one plot of cabbage are sold for 50 pieces of silver.” Today, we would write that with the polynomial 2x + 3y + z = 50. As the transition from writing words to symbols progressed, mathematicians of the day began investigating the properties of these expressions and began to develop better and better ways to solve them leading to the theories and methods we have today.

      Now, perhaps you understand why we put so much emphasis on word problems. This method of codifying problems described by words into a system that permits the easy solution of the problems is so very, very useful. To make this connection, you are asked to translate word problems into math, just as has been done for 100s of years. For example, “I need to build the largest enclosed area I can for my cattle, but I only have 300 meters of fencing material. What should the length of each side be to make this area as big as possible?”

      I hope this helps you start doing the math!
      Keep Studying and Keep Asking Questions!
      (127 votes)
  • leafers seedling style avatar for user Jordyn Robinson
    I get the whole concept in the video really well, however, I had one question:
    In the video, the question states "In the following polynomial", meaning, that there is polynomial involved.

    3x^2 - 8x + 7

    I thought that polynomials were 4+ terms, hence, this will be a trinomial (since it has 3 terms).
    Can anyone explain? :)
    Thank you so much!
    (10 votes)
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  • blobby green style avatar for user blian6833
    How should I learn Algebra 2 to get ahead for next year?
    (11 votes)
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    • leaf green style avatar for user Brooklyn
      Hey! I asked this exact question too before I started. Some things that really helped me were adding Khan's "Algebra 2" course and doing a portion of it everyday throughout the summer. This will keep your mind sharp and teach you the basics of what you'll be doing next year. You can take notes from the videos that you can use later when you are taking the actual class. Make sure to take the mastery quizzes because they are extremely helpful to test your knowledge and teach you what you get wrong if you answer incorrectly. I hope this helps! You got this! Honestly, confidence is everything!
      (19 votes)
  • starky ultimate style avatar for user ▆▇█▓▒░kiwi123321░▒▓█▇▆
    What is coefficient and constant? I get mixed up
    (6 votes)
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    • orange juice squid orange style avatar for user Daberculosis
      The coefficient is the number that is being multiplied by a variable. If you see 12x, the x is the variable and 12 is the coefficient. The prefix co- in front of coefficient means "together". Another word that has co- as a prefix is cooperation. You cooperate when you are working "together" with something or someone else to complete a goal. So just think of coefficient as a number that is cooperating with the variable through multiplication. With 12x, the coefficient 12 is cooperating with variable x.

      A constant is a quantity that does not change it's value. What does constant literally mean? Constant means "remaining the same over time". It doesn't change. It always has the same value. Take the number 6, for example. It's always going to be the value of 6. 6 will never equal 7 or 8 or 9 etc. It has always and will always equal 6. The value is constant, so it is a constant! This is true for any number not connected to a variable. If you look at a variable such as x, it's not a constant, it's a variable. Variables are the opposite of a constant. Variables vary in its value. Variables change depending on the equation and can equal any number.

      Hopefully that clears everything up. Good question.
      (14 votes)
  • starky ultimate style avatar for user Luna Scarlet
    I have to factor (2t^3)-(14t^2)+(24t) using multiplication tables, how do you do that?

    -even my sister who is smarter than me cant figure it out-
    (3 votes)
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    • winston default style avatar for user msalikkhan
      ((2 • (t3)) - (2•7t2)) + 24t >>>
      (2t3 - (2•7t2)) + 24t >>>
      Pull out like factors :
      2t3 - 14t2 + 24t = 2t • (t2 - 7t + 12)>>>
      Factoring t2 - 7t + 12

      The first term is, t2 its coefficient is 1 .
      The middle term is, -7t its coefficient is -7 .
      The last term, "the constant", is +12

      Step-1 : Multiply the coefficient of the first term by the constant 1 • 12 = 12

      Step-2 : Find two factors of 12 whose sum equals the coefficient of the middle term, which is -7 .

      -12 + -1 = -13
      -6 + -2 = -8
      -4 + -3 = -7 That's it

      Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and -3
      t2 - 4t - 3t - 12

      Step-4 : Add up the first 2 terms, pulling out like factors :
      t • (t-4)
      Add up the last 2 terms, pulling out common factors :
      3 • (t-4)
      Step-5 : Add up the four terms of step 4 :
      (t-3) • (t-4)
      Which is the desired factorization

      **Final Answer :- 2t • (t - 3) • (t - 4)**
      (3 votes)
  • aqualine ultimate style avatar for user http://facebookid.khanacademy.org/1414484241
    Does the following equation have 2 terms or 3 terms?
    (2x + 3y) - 7z
    Do the parenthesis around the 2x + 3y make it a single term or is it considered still 2 terms?
    (4 votes)
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  • winston default style avatar for user Grand Theft длgel hellrazor
    Why does a constant have a degree of 0?
    (4 votes)
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  • aqualine ultimate style avatar for user Bloodphionex
    Hi, so my question is if a polynomial has more than 3 terms do you still call it a trinomial or is there a different name for it? or can polynomials not have more than 3 terms? thanks!
    (4 votes)
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    • stelly blue style avatar for user Kim Seidel
      Yes - A polynomial can have more than 3 terms. A trinomial always has 3 terms. A polynomial with more than 3 terms is just a polynomial. If the polynomial with 4 terms, it could be called a quadnomial. But that terminology is rarely used.
      (4 votes)
  • starky sapling style avatar for user phatsolorng
    am currently learning about polynomials what are the rules in adding/subtracting polynomials?
    (2 votes)
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  • piceratops ultimate style avatar for user Lovetocode999
    Would an expression like 9x^2+3x^0 be a polynomial if x=0?
    (5 votes)
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Video transcript

In the following polynomial, identify the terms along with the coefficient and exponent of each term. So the terms are just the things being added up in this polynomial. So the terms here-- let me write the terms here. The first term is 3x squared. The second term it's being added to negative 8x. You might say, hey wait, isn't it minus 8x? And you could just view that as it's being added to negative 8x. So negative 8x is the second term. And then the third term here is 7. It's called a polynomial. Poly, it has many terms. Or you could view each term as a monomial, as a polynomial with only one term in it. So those are the terms. Now let's think about the coefficients of each of the terms. The coefficient is what's multiplying the power of x or what's multiplying in the x part of the term. So over here, the x part is x squared. That's being multiplied by 3. So 3 is the coefficient on the first term. On the second term, we have negative 8 multiplying x. And we want to be clear, the coefficient isn't just 8. It's a negative 8. It's negative 8 that's multiplying x. So that's the coefficient right over here. And here you might say, hey wait, nothing is multiplying x here. I just have a 7. There is no x. 7 isn't being multiplied by x. But you can think of this as 7 being multiplied by x to the 0 because we know that x to the zeroth power is equal to 1. So we would even call this constant, the 7, this would be the coefficient on 7x to the 0. So you could view this as a coefficient. So this is also a coefficient. So let me make it clear, these three things are coefficients. Now the last part, they want us to identify the exponent of each term. So the exponent of this first term is 2. It's being raised to the second power. The exponent of the second term, remember, negative 8x, x is the same thing as x to the first power. So the exponent here is 1. And then on this last term, we already said, 7 is the same thing as 7x to the 0. So the exponent here on the constant term on 7 is 0. So these things right over here, those are our exponents. And we are done.