Polynomials Basic Vocabulary Variable a letter that represents

Polynomials

Basic Vocabulary • Variable: a letter that represents an unknown value • Expression: a group of terms written without an equal sign. • Polynomial: The sum or difference of many expressions • Term: Each piece or part of the polynomial that is being added or subtracted. • Exponent: the power to which a term or variable is raised. • Coefficient: the number in front of a variable. • Degree: the exponent of a term.

• The variables in the terms of any polynomial must be raised to a whole number; that means no square roots, no fraction exponents, and no negative exponents. • Also, there must not be any variables in the denominator. Polynomial s

Examples of a Polynomials • Here are some examples: Example Polynomial or Not Reason why not a polynomial 6 x – 2 NOT a polynomial term This has a negative exponent. 1/ x 2 NOT a polynomial term This has the variable in the denominator. sqrt(x) NOT a polynomial term This has the variable inside a radical. 4 x 2 a polynomial term

Typical Polynomial • The first term has exponent 2; • the second term has an understood exponent 1; • and the last term doesn't have any variable at all, meaning the exponent is 0, ( x 0 = 1).

Order of a Polynomial • Polynomials are usually written with the terms written in "decreasing" order; that is, with the highest exponent first, the next highest next, and so forth, until you get down to the plain old number. • The first term in the polynomial, when it is written in decreasing order, is also the term with the biggest exponent, and is called the "leading term".

Degree of a Polynomial • The exponent on a term tells you the "degree" of the term. • The degree of the leading term tells you the degree of the whole polynomial;

Examples • 2 x 5 – 5 x 3 – 10 x + 9 Degree 5 • 7 x 4 + 6 x 2 + x Degree 4

Coefficient vs. Leading Coefficient • When a term contains both a number and a variable part, the number part is called the "coefficient". The coefficient on the leading term is called the leading coefficient.

Polynomial Breakdown • The "poly" in "polynomial" means "many". • a one-term polynomial, such as 2 x or 4 x 2, may be called a "monomial" • a two-term polynomial, such as 2 x + y or x 2 – 4, may be called a "binomial" • a three-term polynomial, such as 2 x + y + z or x 4 + 4 x 2 – 4, may be called a "trinomial"

Names of Common Polynomials (by their degrees) • Linear—any polynomial with a degree of one. Polynomial Name • Quadratic—any polynomial with a Linear degree of two. • Cubic—any polynomial with a degree of three. Quadrati c Cubic Degree Example 1 2 x + 4 2 2 x 2 + 4 3 2 x 3 + 4

Evaluation • "Evaluating" a polynomial is the same as evaluating anything else: you plug in the given value of x, and figure out what the answer will be, or, in some cases, what y is supposed to be. For instance: • Evaluate 2 x 3 – x 2 – 4 x + 2 at x = – 3 Plug in – 3 for x, remembering to be careful with parentheses and negatives: 2(– 3)3 – (– 3)2 – 4(– 3) + 2 = 2(– 27) – (9) + 12 + 2 = – 54 – 9 + 14 = – 63 + 14 = – 49 Always remember to be careful with the minus signs!

Rules for Combining Like Terms in Polynomials 4 x and 3 NOT like terms The second term has no variable The second term now has a variable, but it doesn't match the variable of the first term 4 x and 3 y NOT like terms 4 x and 3 x 2 The second term now has the same NOT like terms variable, but the degree is different 4 x and 3 x LIKE TERMS Now the variables match and the degrees match

Examples Simplify 3 x + 4 x 3 x + 4 x = 7 x l Simplify 2 x 2 + 3 x – 4 – x 2 + x + 9 = (2 x 2 – x 2)+ (3 x + x) + (– 4 + 9) = x 2 + 4 x + 5 l

Examples Continued Simplify 10 x 3 – 14 x 2 + 3 x – 4 x 3 + 4 x – 6 = (10 x 3 – 4 x 3) + (– 14 x 2) + (3 x + 4 x) – 6 = 6 x 3 – 14 x 2 + 7 x – 6 n Simplify 25 – (x + 3 – x 2) = 25 – x – 3 + x 2 = x 2 – x + 25 – 3 = x 2 – x + 22 n

Multiplication vs. Addition • x + x = 2 x • x * x = x 2 Notice 2 x ≠ x 2 • So, if you have something like x 3 + x 2, DO NOT say that this somehow equals something like x 5 or 5 x. • If you have something like 2 x + x, DO NOT say that this somehow equals something like 2 x 2.

Addition/Subtraction • When adding or subtracting Polynomials, you can only add like terms (terms that have the same variable and same degree). • The sum or difference will have the same variable and degree, but the only thing that will change is the coefficient. • Ex: 3 x 2 – 3 x + 7 x + 5 = 3 x 2 + 4 x + 5

Multiplication § When multiplying terms, you must add the exponents of the like variables and multiply any coefficients like you would normally do. § Ex: 2 xy(5 x) = 2*5 * y * x*x = 10* y * x 2 = 10 yx 2

Division o When dividing polynomials, you must subtract the bottom power from the top power and divide the coefficients like normal.