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:
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6. \( h(x) \):
- Polynomial: Yes
- Degree: 4
- Type: Quartic
- Leading Coefficient: -3
- Constant Term: -6
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7. \( f(x) \):
- Polynomial: Yes
- Degree: 4
- Type: Quartic
- Leading Coefficient: 7
- Constant Term: -26
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8. \( g(x) \):
- Polynomial: Yes
- Degree: 6
- Type: Sextic
- Leading Coefficient: 1
- Constant Term: -51