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Chapter 4
 

Chapter 4

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    Chapter 4 Chapter 4 Presentation Transcript

    • MTH 209, Chapter 4 slides By Marie E. Borrazzo
    • What’s a Polynomial?
      • Polynomials are sums of "variable and exponent" expressions.
      • Each piece of the polynomial, each part that is being added, is called a "term".
      • Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions.
    • What’s a Polynomial?
      • Poly - nomials
      • Many names
    • Here are some examples:
        • 4x 4 + 3y 2
        • 3x 2 + 4x + 3
        • 3r -5
    • The Parts of a Polynomial
      • 3x 2 + 4x + 6
      Leading coefficient Constant
    • The degree of a Polynomial
      • 3x 2 + 4x + 6
      The degree of the polynomial is determined by the highest exponent power of the polynomial. This is a second degree polynomial.
    • The terms of a Polynomial
      • 3x 2 + 4x + 6
      This polynomial has three terms : 3x 2 , 4x, and 6
    • Another example
      • Give the degree of the following polynomial: 2x 5 – 5x 3 – 10x + 9
      • This polynomial has four terms, including a fifth-degree term, a third-degree term, a first-degree term, and a constant term.
    • Degrees and terms?
      • Give the degree of the following polynomial: 7x 4 + 6x 2 + x
      • This polynomial has three terms, including a fourth-degree term, a second-degree term, and a first-degree term. There is no constant term. This is a fourth-degree polynomial.
    • 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.
    • Monomial
      • a one-term polynomial, such as 2x or 4x 2 , may also be called a "monomial" ("mono" meaning "one")
    • Binomial
      • a two-term polynomial, such as 2x + y or x 2 – 4, may also be called a "binomial" ("bi" meaning "two")
    • Trinomial
      • a three-term polynomial, such as 2x + y + z or x 4 + 4x 2 – 4, may also be called a "trinomial" ("tri" meaning "three")
    • Polynomials and their degree:
      • a second-degree polynomial, such as 4x 2 , x 2 – 9, or ax 2 + bx + c, is also called a "quadratic"
      • a third-degree polynomial, such as –6x 3 or x 3 – 27, is also called a "cubic"
      • a fourth-degree polynomial, such as x 4 or 2x 4 – 3x 2 + 9, is sometimes called a "quartic"
      • a fifth-degree polynomial, such as 2x 5 or x 5 – 4x 3 – x + 7, is sometimes called a "quintic"
    • Adding polynomials
      • When you add polynomials you will merely worry about the coefficient not the variable and its exponent.
      • Remember you can ONLY add like terms.
    • Adding polynomials
      • Lets add monomials: 2x 2 and 3x 2
      • 2 x 2 + 3 x 2 = 5 x 2
      • Notice the x square remains unchanged.
      • If these terms had different exponent values you could NOT add or subtract them .
    • Subtracting polynomials
      • Lets subtract: 7x 2 and 3x 2
      • 7 x 2 - 3 x 2 = 4 x 2
      • Notice the x square remains unchanged.
      • If these terms had different exponent values you could NOT add or subtract them .
    • Adding polynomials
      • Lets add a trinomial and a binomial:
      • 7x 2 - 4x + 4
      • 3x 2 + 6
      • 7x 2 - 4x + 4 plus 3x 2 + 6 = 10x 2 - 4x+ 10
      • Notice the exponents remain unchanged.
      • Notice you can only add like terms.
    • Subtracting polynomials
      • Lets subtract a trinomial and a binomial:
      • 7x 2 - 4x + 4
      • 3x 2 + 6
      • 7x 2 - 4x + 4 minus ( 3x 2 + 6) = 4 x 2 - 4x - 2
      • Notice the exponents remain unchanged.
      • Notice you can only add like terms.