Sure! Let's break down the given polynomial [tex]\(\left(\frac{x^3 + 2x + 1}{5}\right) - \left(\frac{7}{2}\right)x^2 - x^6\)[/tex] and extract the required information step-by-step.
### Part (i) The degree of the polynomial
The degree of a polynomial is the highest power of [tex]\(x\)[/tex] in the polynomial. Let's analyze the given polynomial:
[tex]\[ \left(\frac{x^3 + 2x + 1}{5}\right) - \left(\frac{7}{2}\right)x^2 - x^6 \][/tex]
Here, the term with the highest power of [tex]\(x\)[/tex] is [tex]\(-x^6\)[/tex]. The highest power is 6. Therefore, the degree of the polynomial is:
[tex]\[
\boxed{6}
\][/tex]
### Part (ii) The coefficient of [tex]\( x^3 \)[/tex]
Next, we need to find the coefficient of [tex]\(x^3\)[/tex] in the polynomial:
[tex]\[ \left(\frac{x^3 + 2x + 1}{5}\right) - \left(\frac{7}{2}\right)x^2 - x^6 \][/tex]
First, consider the term [tex]\(\frac{x^3 + 2x + 1}{5}\)[/tex]. When we expand this term, we get:
[tex]\[ \frac{x^3}{5} + \frac{2x}{5} + \frac{1}{5} \][/tex]
From this expansion, the coefficient of [tex]\(x^3\)[/tex] is [tex]\(\frac{1}{5}\)[/tex] or 0.2.
[tex]\[
\boxed{0.2}
\][/tex]
### Part (iii) The coefficient of [tex]\( x^6 \)[/tex]
Now, we need to find the coefficient of [tex]\(x^6\)[/tex]. Looking at the polynomial:
[tex]\[ \left(\frac{x^3 + 2x + 1}{5}\right) - \left(\frac{7}{2}\right)x^2 - x^6 \][/tex]
We see that the only term containing [tex]\(x^6\)[/tex] is [tex]\(-x^6\)[/tex]. The coefficient of this term is [tex]\(-1\)[/tex].
[tex]\[
\boxed{-1}
\][/tex]
### Part (iv) The constant term
Finally, we need to determine the constant term in the polynomial. Let's inspect the term [tex]\(\frac{x^3 + 2x + 1}{5}\)[/tex] again. As we already expanded it as:
[tex]\[ \frac{x^3}{5} + \frac{2x}{5} + \frac{1}{5} \][/tex]
The constant term in this expansion is [tex]\(\frac{1}{5}\)[/tex] or 0.2. Since there are no other constant terms in the polynomial, this is the constant term of the entire polynomial.
[tex]\[
\boxed{0.2}
\][/tex]
### Summary:
(i) The degree of the polynomial is [tex]\( \boxed{6} \)[/tex].
(ii) The coefficient of [tex]\( x^3 \)[/tex] is [tex]\( \boxed{0.2} \)[/tex].
(iii) The coefficient of [tex]\( x^6 \)[/tex] is [tex]\( \boxed{-1} \)[/tex].
(iv) The constant term is [tex]\( \boxed{0.2} \)[/tex].
