Question

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


1. F(x) = - 3x + 5x³-6x²+2

2. p(x)=1 half x²+ 3x-4x³+6x⁴-1

3. f(x)=9x⁴+8x³-6x–²+2x

4. g(x)=√3 - 12x + 13x²

5. h(x)=5 3rds x²-√7x⁴+8x³- 1 half + x

Answer 1

To determine whether a function is a polynomial, we check if it consists solely of non-negative integer powers of the variable and coefficients that are real numbers.

Let's analyze each given function:

1. **\( F(x) = -3x + 5x^3 - 6x^2 + 2 \)**
- Yes, this is a polynomial function.
- Standard form: \( 5x^3 - 6x^2 - 3x + 2 \)
- Degree: 3 (the highest power of \( x \))
- Type: Cubic polynomial
- Leading coefficient: 5 (the coefficient of the term with the highest degree)
- Constant term: 2

2. **\( p(x) = \frac{1}{2}x^2 + 3x - 4x^3 + 6x^4 - 1 \)**
- Yes, this is a polynomial function.
- Standard form: \( 6x^4 - 4x^3 + \frac{1}{2}x^2 + 3x - 1 \)
- Degree: 4
- Type: Quartic polynomial
- Leading coefficient: 6
- Constant term: -1

3. **\( f(x) = 9x^4 + 8x^3 - 6x^{-2} + 2x \)**
- No, this is not a polynomial function due to the presence of the term \( -6x^{-2} \) (which has a negative exponent).

4. **\( g(x) = \sqrt{3} - 12x + 13x^2 \)**
- Yes, this is a polynomial function.
- Standard form: \( 13x^2 - 12x + \sqrt{3} \)
- Degree: 2
- Type: Quadratic polynomial
- Leading coefficient: 13
- Constant term: \( \sqrt{3} \)

5. **\( h(x) = \frac{5}{3}x^2 - \sqrt{7}x^4 + 8x^3 - \frac{1}{2} + x \)**
- Yes, this is a polynomial function.
- Standard form: \( -\sqrt{7}x^4 + 8x^3 + \frac{5}{3}x^2 + x - \frac{1}{2} \)
- Degree: 4
- Type: Quartic polynomial
- Leading coefficient: -\( \sqrt{7} \)
- Constant term: -\( \frac{1}{2} \)

### Summary

1. **F(x)**: Polynomial, Degree 3, Leading Coefficient 5, Constant Term 2
2. **p(x)**: Polynomial, Degree 4, Leading Coefficient 6, Constant Term -1
3. **f(x)**: Not a polynomial
4. **g(x)**: Polynomial, Degree 2, Leading Coefficient 13, Constant Term \( \sqrt{3} \)
5. **h(x)**: Polynomial, Degree 4, Leading Coefficient -\( \sqrt{7} \), Constant Term -\( \frac{1}{2} \)