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### Polynomial

• 2. Polynomial comes from poly- (meaning "many") and - nomial (in this case meaning "term") ... so it says "many terms" A polynomial can have: constants (like 3, -20, or ½) variables (like x and y) exponents (like the 2 in y2), but only 0, 1, 2, 3, ... etc are allowed
• 4. These are polynomials • 3x • x - 2 • -6y2 - (7/9)x • 3xyz + 3xy2z - 0.1xz - 200y + 0.5 • 512v5+ 99w5 • 5 (Yes, even "5" is a polynomial, one term is allowed, and it can even be just a constant!)
• 5. And these are not polynomials • 2/(x+2) is not, because dividing by a variable is not allowed • 1/x is not either • 3xy-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...) • √x is not, because the exponent is "½" (see fractional exponents)
• 6. • x/2 is allowed, because you can divide by a constant • also 3x/8 for the same reason • √2 is allowed, because it is a constant (= 1.4142...etc) But these are allowed:
• 7. Monomial, Binomial, Trinomial • (There is also quadrinomial (4 terms) and quintinomial (5 terms), but those names are not often used) How do you remember the names? Think cycles!
• 8. • Polynomials can have no variable at all • Example: 21 is a polynomial. It has just one term, which is a constant. • Or one variable • Example: x4-2x2+x has three terms, but only one variable (x) • Or two or more variables • Example: xy4-5x2z has two terms, and three variables (x, y and z) Variables
• 9. • Because of the strict definition, polynomials are easy to work with. • For example we know that: • If you add polynomials you get a polynomial • If you multiply polynomials you get a polynomial • So you can do lots of additions and multiplications, and still have a polynomial as the result. • Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines. Example: x4-2x2+x What is Special About Polynomials?
• 10. Degree • The degree of a polynomial with only one variable is the largest exponent of that variable. • Example: • The Degree is 3 (the largest exponent of  x) • For more complicated cases, read Degree (of an Expression).
• 11. Standard Form • The Standard Form for writing a  polynomial is to put the terms with the  highest degree first. • Example: Put this in Standard Form:  3x2 - 7 + 4x3 + x6 • The highest degree is 6, so that goes  first, then 3, 2 and then the constant  last: • x6 + 4x3 + 3x2 - 7
• 12. Like Terms • are terms whose variables (and  their exponents such as the 2 in x2) are the  same. • In other words, terms that are "like" each  other. • Note: the coefficients (the numbers you  multiply by, such as "5" in 5x) can be  different. • Example: • 7x x -2x πx • are all like terms because the variables are  all x