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# Geometry/Polynomials

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Hello sir

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Hi Aishwarya,

1) Think about anything you have ever had to calculate such as tips, tax, or how much change to return.  These are simple uses of algebra, there are many more.

2) The analogy is not perfect.  My point is that "expression" is a word that means "one or more terms".  We use it in a general sense.  For instance, you could have a problem like this:

Simplify the expressions:
a) (2x+2y)(2x)
b) 2x2y
c) 2x(-5x+3)

I honestly hope this clears it up.  Again, this is not something you should have to worry about.  If you can present me a concrete example of you having lost marks because of this distinction, I would love to see it to be able to assist you further.

3)  A constant is a number.  It does not change values from one problem to the next.  x and y can take on different values from problem to problem.  2 and -90 cannot.  Even if not fully simplified, a constant is still a constant.  12+4=16, a constant.

4) As far as I know, it's just for simplicity.  The word "binomial" is used quite often in algebra.  "Monomial" and "trinomial" not so much.  "Polynomial" is the catch-all word.

5) Polynomials have a degree (I will explain polynomials below).  If an equation can be rearranged into 0=[polynomial], then the degree of that equation is the highest power in the polynomial.  The degree of the polynomial gives it certain properties.  Unfortunately the truth is most applications of degree will only be seen in university-level mathematics.  For now, you should be able to determine the degree of a polynomial, and to be able to recognize polynomials of degree 0, 1, and 2.

6) Not necessarily.  A monomial is a product of powers of variables.  For example 7xy˛.  This could instead be written as (7x^0)(x^1)(y^2).  x and y are variables, so what we have is a product of powers of variables.  The powers must be non-negative integers--more on this below.
Examples of terms that are not monomials are 2^x, x^0.5, or more exotic things like sin(x) or log(x).
However, your question was about an expression in one variable, which can fail to be a monomial in an even simpler way.  x+1 is an expression in one variable, but it is actually a binomial.

7) I have defined monomial above.  A polynomial is a sum of one or more monomials.  It turns out that when we restrict the exponents to non-negative integers, many special properties emerge.  Again, you probably will not deal too heavily with any of these until university-level courses such as calculus.  For now, being able to recognize a polynomial should be sufficient.

Thanks for asking,
Azeem

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#### Azeem Hussain

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I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

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4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

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CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

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Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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