Question

use the function $f(x) = 2x^3 + 4x^2 + 5x - 7$ to complete the following sentences. the degree is \boxed{}. the leading coefficient is \boxed{}. the value of the function when $x = 0$ is \boxed{}.

Explanation:

Step1: Determine the degree of the polynomial

The degree of a polynomial is the highest power of \( x \) in the polynomial. For \( f(x) = 2x^3 + 4x^2 + 5x - 7 \), the powers of \( x \) are 3, 2, 1, and 0 (for the constant term -7). The highest power is 3.

Step2: Determine the leading coefficient

The leading coefficient is the coefficient of the term with the highest degree. The term with the highest degree (\( x^3 \)) is \( 2x^3 \), so the leading coefficient is 2.

Step3: Find the value of the function when \( x = 0 \)

Substitute \( x = 0 \) into \( f(x) \): \( f(0) = 2(0)^3 + 4(0)^2 + 5(0) - 7 \). Calculating each term: \( 2(0) = 0 \), \( 4(0) = 0 \), \( 5(0) = 0 \), so \( f(0) = -7 \).

Answer:

The degree is \( 3 \).
The leading coefficient is \( 2 \).
The value of the function when \( x = 0 \) is \( -7 \).