Certainly! Let’s analyze the polynomial given:
[tex]\[ \frac{x^3 + 2x + 1}{5} - \frac{7}{2}x^2 - x^6 \][/tex]
To begin, let's rewrite the polynomial by distributing the division by 5 across the terms inside the parentheses:
[tex]\[
\frac{x^3}{5} + \frac{2x}{5} + \frac{1}{5} - \frac{7}{2}x^2 - x^6
\][/tex]
Now, our polynomial appears as:
[tex]\[
-\frac{7}{2}x^2 - x^6 + \frac{x^3}{5} + \frac{2x}{5} + \frac{1}{5}
\][/tex]
Given this polynomial, let's identify each part required:
### (i) Degree of the polynomial
The degree of a polynomial is the highest power of \( x \) with a non-zero coefficient. Here, the term with the highest power is \( -x^6 \). Therefore, the degree of the polynomial is:
[tex]\[
6
\][/tex]
### (ii) Coefficient of \( x^3 \)
The term involving \( x^3 \) is \(\frac{x^3}{5}\). The coefficient of this term is:
[tex]\[
\frac{1}{5} = 0.2
\][/tex]
### (iii) Coefficient of \( x^6 \)
The term involving \( x^6 \) is \( -x^6 \). The coefficient of this term is:
[tex]\[
-1
\][/tex]
### (iv) Constant term
The constant term in the polynomial is the term that does not contain any \( x \). Here, it is:
[tex]\[
\frac{1}{5} = 0.2
\][/tex]
### Summary
(i) The degree of the polynomial is \( 6 \).
(ii) The coefficient of \( x^3 \) is \( 0.2 \).
(iii) The coefficient of \( x^6 \) is \( -1 \).
(iv) The constant term is [tex]\( 0.2 \)[/tex].
