Question

Is it degree, type, leading coefficient, and constant term h(x)=-3x⁴+2x³-12x-6;x=-2

Answer 1

To analyze the polynomial \( h(x) = -3x^4 + 2x^3 - 12x - 6 \):

1. **Degree**: The degree of a polynomial is the highest exponent of \( x \). In this case, the highest exponent is 4 (from the term \( -3x^4 \)), so the degree of the polynomial is **4**.

2. **Type**: This polynomial is a **quartic polynomial** because its degree is 4.

3. **Leading Coefficient**: The leading coefficient is the coefficient of the term with the highest degree. Here, the term with the highest degree is \( -3x^4 \), so the leading coefficient is **-3**.

4. **Constant Term**: The constant term is the term without \( x \), which is \( -6 \). Therefore, the constant term is **-6**.

Now, let's evaluate \( h(x) \) at \( x = -2 \):

\[
h(-2) = -3(-2)^4 + 2(-2)^3 - 12(-2) - 6
\]

Calculating each term step by step:

- \( (-2)^4 = 16 \) so \( -3(-2)^4 = -3 \times 16 = -48 \)
- \( (-2)^3 = -8 \) so \( 2(-2)^3 = 2 \times -8 = -16 \)
- \( -12(-2) = 24 \)
- The constant term is \( -6 \).

Now put it all together:

\[
h(-2) = -48 - 16 + 24 - 6
\]

Calculating it step-by-step:

\[
h(-2) = -48 - 16 = -64
\]
\[
-64 + 24 = -40
\]
\[
-40 - 6 = -46
\]

Thus, \( h(-2) = -46 \).

In summary:
- Degree: **4**
- Type: **Quartic**
- Leading Coefficient: **-3**
- Constant Term: **-6**
- \( h(-2) = -46 \)