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Polynomials and Radical Expressions

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degree (of a monomial) = the sum of the exponents of the variables. ... If a variable is under a radical sign or in a denominator, then it is not a monomial. ... – PowerPoint PPT presentation

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Title: Polynomials and Radical Expressions


1
Polynomials and Radical Expressions
  • (Chapter 5)

2
Properties of Exponents
  • am . an a(m n)
  • am / an a(m ? n)
  • (am)n a(m?n)
  • (a ? b)m am ? bm
  • (a / b)m am / bm
  • a0 1

3
Negative Exponents
  • a-n 1 and 1 an an a-n
  • Scientific Notation
  • a x 10n
  • where 1 lt a lt 10 and n is an integer.

4
Monomials (5.1)
  • monomial an expression that is a number, a
    variable, or the product of a number and one or
    more variables.
  • constant a monomial that contains no variables.

5
  • coefficient the number part of a monomial.
  • degree (of a monomial) the sum of the exponents
    of the variables.
  • 3 is a constant (degree of 0).
  • 3x2 has a degree of 2.
  • 3x2y4z5 has a degree of 11.

6
  • Variables in monomials must have whole number
    exponents.
  • If a variable is under a radical sign or in a
    denominator, then it is not a monomial.
  • and 1/x2 are not monomials.

7
Polynomials (5.3)
  • polynomial a monomial or a sum of monomials.
  • terms the monomials that make up a polynomial.
  • like terms terms with the same variables with
    the same exponents.

8
  • binomial a polynomial with two unlike terms.
  • trinomial a polynomial with three unlike terms.
  • degree (of a polynomial) the degree of its
    largest monomial.

9
Adding Polynomials
  • To add polynomials, add like terms.
  • To subtract polynomials, subtract like terms, or
  • distribute the negative into the second
    polynomial and then add like terms.

10
Multiplying Polynomials
  • To multiply monomials, multiply the coefficients
    and follow the exponent rules for the variables.
  • To multiply two binomials together, use F.O.I.L.
  • To multiply larger polynomials, use M.E.T.W.E.T.

11
Dividing Polynomials (5.3)
  • To divide a polynomial by a monomial, divide each
    term by the monomial.
  • To divide a polynomial by another polynomial,
    use long division.
  • (D.M.S.B.R.)

12
  • To divide a polynomial by a first-degree
    binomial, use synthetic division.
  • Write the coefficients in a row (in descending
    order).
  • Write the opposite of the constant of the divisor
    to the left.
  • Bring down the first number.

13
  • Multiply the number with the constant.
  • Add the product with the next coefficient.
  • Repeat steps 4 and 5 until there are no more
    numbers.
  • Write the last answer as a remainder.

14
  • If the divisor has a leading coefficient other
    than 1, that must be divided separately.
  • Divide it out of the divisor first.
  • Do the synthetic division.
  • Then divide it out of the final answer.

15
Factoring (5.4)
  • factors (of a polynomial) polynomials that
    divide evenly into a polynomial.
  • prime polynomials polynomials that cannot be
    factored.

16
Types of Factoring
  • Common factors
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
  • Perfect Square Trinomials
  • Guess and Check
  • Grouping (for 4 or more terms)

17
Graphs of Polynomial Functions (5.2)
  • A polynomial function cannot have
  • Negative exponents on the variables
  • Variables in the denominator
  • Fractional exponents on the variables
  • Variables in a radicand

18
End Behavior of Polynomials
  • To determine the shape of a graph of a polynomial
    function, look at its leading coefficient and its
    degree.
  • PEHH POLH NELL NOHL
  • (The x-intercepts and y-intercept will also be
    important in graphing polynomial functions.)
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