Question
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
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
Answers
GPT-4o mini
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
### 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