Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To find all the real zeros of the given polynomial and write the polynomial in factored form, let's break it down step-by-step.
We are given the polynomial:
[tex]\[ P(x) = x^4 - 2x^3 - 48x^2 + 98x - 49 \][/tex]
### Step 1: Find the Zeros
The zeros are the values of [tex]\( x \)[/tex] that make the polynomial equal to zero. According to our factorization:
[tex]\[ P(x) = (x - 7)(x - 1)^2(x + 7) \][/tex]
By setting each factor equal to zero and solving for [tex]\( x \)[/tex], we get the zeros of the polynomial. Let's solve it:
1. [tex]\((x - 7) = 0 \quad \Rightarrow \quad x = 7\)[/tex]
2. [tex]\((x - 1)^2 = 0 \quad \Rightarrow \quad x = 1\)[/tex] (Note: It's a double root because of the squared term)
3. [tex]\((x + 7) = 0 \quad \Rightarrow \quad x = -7\)[/tex]
So, the zeros of the polynomial [tex]\( P(x) \)[/tex] are [tex]\( x = 7 \)[/tex], [tex]\( x = 1 \)[/tex] (with multiplicity 2), and [tex]\( x = -7 \)[/tex].
### Step 2: Write the Polynomial in Factored Form
The factored form combines all the found zeros into the expression based on their multiplicities:
[tex]\[ P(x) = (x - 7)(x - 1)^2(x + 7) \][/tex]
### Summary
The real zeros of the polynomial [tex]\( P(x) = x^4 - 2x^3 - 48x^2 + 98x - 49 \)[/tex] are [tex]\( x = 7 \)[/tex], [tex]\( x = 1 \)[/tex] (with multiplicity 2), and [tex]\( x = -7 \)[/tex].
In factored form, the polynomial is:
[tex]\[ P(x) = (x - 7)(x - 1)^2(x + 7) \][/tex]
Thus, the final answer can be rewritten as follows:
[tex]\[ \boxed{ \begin{array}{l} \text{Zeros: } x = 7, \, x = 1 \, (\text{multiplicity 2}), \, x = -7 \\ \text{Factored Polynomial: } P(x) = (x - 7)(x - 1)^2(x + 7) \end{array} } \][/tex]
We are given the polynomial:
[tex]\[ P(x) = x^4 - 2x^3 - 48x^2 + 98x - 49 \][/tex]
### Step 1: Find the Zeros
The zeros are the values of [tex]\( x \)[/tex] that make the polynomial equal to zero. According to our factorization:
[tex]\[ P(x) = (x - 7)(x - 1)^2(x + 7) \][/tex]
By setting each factor equal to zero and solving for [tex]\( x \)[/tex], we get the zeros of the polynomial. Let's solve it:
1. [tex]\((x - 7) = 0 \quad \Rightarrow \quad x = 7\)[/tex]
2. [tex]\((x - 1)^2 = 0 \quad \Rightarrow \quad x = 1\)[/tex] (Note: It's a double root because of the squared term)
3. [tex]\((x + 7) = 0 \quad \Rightarrow \quad x = -7\)[/tex]
So, the zeros of the polynomial [tex]\( P(x) \)[/tex] are [tex]\( x = 7 \)[/tex], [tex]\( x = 1 \)[/tex] (with multiplicity 2), and [tex]\( x = -7 \)[/tex].
### Step 2: Write the Polynomial in Factored Form
The factored form combines all the found zeros into the expression based on their multiplicities:
[tex]\[ P(x) = (x - 7)(x - 1)^2(x + 7) \][/tex]
### Summary
The real zeros of the polynomial [tex]\( P(x) = x^4 - 2x^3 - 48x^2 + 98x - 49 \)[/tex] are [tex]\( x = 7 \)[/tex], [tex]\( x = 1 \)[/tex] (with multiplicity 2), and [tex]\( x = -7 \)[/tex].
In factored form, the polynomial is:
[tex]\[ P(x) = (x - 7)(x - 1)^2(x + 7) \][/tex]
Thus, the final answer can be rewritten as follows:
[tex]\[ \boxed{ \begin{array}{l} \text{Zeros: } x = 7, \, x = 1 \, (\text{multiplicity 2}), \, x = -7 \\ \text{Factored Polynomial: } P(x) = (x - 7)(x - 1)^2(x + 7) \end{array} } \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.