Determine whether the function is a polynomial function. If so, write it in standard form and state its degree, type, leading coefficient, and constant term

6. h(x) = - 3x⁴+2x³-12x-6;x=-2

7. f(x) = 7x⁴-10x²+14 x -26;x=-7

8. g(x) = x⁶-64x⁴+x²-7x-51;x=8

1 answer

To determine whether each function is a polynomial function, we will check if it consists of terms in the form \( ax^n \), where \( a \) is a real number and \( n \) is a non-negative integer. If it is a polynomial, we will write it in standard form (terms arranged by descending powers of \( x \)), state its degree (the highest power of \( x \)), type (which could be linear, quadratic, cubic, quartic, etc.), leading coefficient (the coefficient of the term with the highest degree), and constant term (the term with \( x^0 \)).

6. \( h(x) = -3x^4 + 2x^3 - 12x - 6 \)

  • Polynomial Function: Yes, it is a polynomial because all the terms are of the form \( ax^n \) with non-negative integer exponents.
  • Standard Form: It is already in standard form.
  • Degree: 4 (the highest power of \( x \) is 4).
  • Type: Quartic (since the degree is 4).
  • Leading Coefficient: -3 (the coefficient of the \( x^4 \) term).
  • Constant Term: -6 (the term without \( x \)).

7. \( f(x) = 7x^4 - 10x^2 + 14x - 26 \)

  • Polynomial Function: Yes, it is a polynomial.
  • Standard Form: It is already in standard form.
  • Degree: 4 (the highest power of \( x \) is 4).
  • Type: Quartic.
  • Leading Coefficient: 7 (the coefficient of the \( x^4 \) term).
  • Constant Term: -26 (the term without \( x \)).

8. \( g(x) = x^6 - 64x^4 + x^2 - 7x - 51 \)

  • Polynomial Function: Yes, it is a polynomial.
  • Standard Form: It is already in standard form.
  • Degree: 6 (the highest power of \( x \) is 6).
  • Type: Sextic.
  • Leading Coefficient: 1 (the coefficient of the \( x^6 \) term).
  • Constant Term: -51 (the term without \( x \)).

Summary:

  • 6. \( h(x) \):

    • Polynomial: Yes
    • Degree: 4
    • Type: Quartic
    • Leading Coefficient: -3
    • Constant Term: -6
  • 7. \( f(x) \):

    • Polynomial: Yes
    • Degree: 4
    • Type: Quartic
    • Leading Coefficient: 7
    • Constant Term: -26
  • 8. \( g(x) \):

    • Polynomial: Yes
    • Degree: 6
    • Type: Sextic
    • Leading Coefficient: 1
    • Constant Term: -51