Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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].
### 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].
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
