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
• \( h(-2) = -46 \)