Polynomials and Radical Expressions

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Polynomials and Radical Expressions

• (Chapter 5)

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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

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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.

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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.

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• 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.

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• 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.

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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.

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

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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.

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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.

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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.)

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• 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.

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• 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.

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• 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.

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Factoring (5.4)

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

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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)

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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

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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.)