Question

which of the following statements is true of the polynomial $f(x)=x^3 - 6x^2 - 4 + 2x^5$?

a. the degree of the polynomial is 3

b. the standard form of the polynomial is $f(x)=2x^5 - 6x^2 + x^3 - 4$

c. the leading coefficient of the polynomial is 1.

d. the polynomial is a trinomial

Explanation:

Step1: Recall Polynomial Standard Form

A polynomial in standard form is written with terms in descending order of degree. The general form is \( a_nx^n + a_{n - 1}x^{n - 1}+\dots+a_1x + a_0 \), where \( a_n
eq0 \) and \( n \) is the highest degree (exponent) of \( x \).

Step2: Analyze Given Polynomial \( f(x)=x^2 - 6x^3 - 4 + 2x^5 \)

• Identify the degrees of each term:
• For \( x^2 \), the degree is \( 2 \).
• For \( -6x^3 \), the degree is \( 3 \).
• For \( -4 \), the degree is \( 0 \) (constant term, since \( x^0 = 1 \), so \( -4=-4x^0 \)).
• For \( 2x^5 \), the degree is \( 5 \).
• Rearrange the terms in descending order of their degrees:
• The term with degree \( 5 \): \( 2x^5 \)
• The term with degree \( 3 \): \( -6x^3 \)
• The term with degree \( 2 \): \( x^2 \)
• The constant term (degree \( 0 \)): \( -4 \)

So, the standard form is \( f(x)=2x^5 - 6x^3 + x^2 - 4 \), which matches option B.

Step3: Analyze Other Options (to Confirm)

• Option A: The degree of a polynomial is the highest power of \( x \). Here, the highest power is \( 5 \) (from \( 2x^5 \)), so the degree is \( 5 \), not \( 3 \). Eliminate A.
• Option C: The leading coefficient is the coefficient of the term with the highest degree. For \( 2x^5 \), the coefficient is \( 2 \), not \( 1 \). Eliminate C.
• Option D: A binomial has exactly two terms. The given polynomial has four terms (\( 2x^5, -6x^3, x^2, -4 \)), so it is not a binomial. Eliminate D.

Answer:

B. The standard form of the polynomial is \( f(x) = 2x^5 - 6x^3 + x^2 - 4 \)