Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
Sure! Let's tackle this step-by-step.
### Given:
[tex]\[ f(x) = 9x^3 - 97x^2 + 304x - 182 \][/tex]
A known zero is [tex]\(5 + i\)[/tex].
### Step 1: Identify the Conjugate Zero
Since the coefficients of the polynomial are real, the complex conjugate of any complex zero is also a zero. Therefore, if [tex]\(5 + i\)[/tex] is a zero, [tex]\(5 - i\)[/tex] must also be a zero.
### Step 2: Form the Quadratic Factor
The zeros [tex]\(5 + i\)[/tex] and [tex]\(5 - i\)[/tex] imply a quadratic factor of the polynomial [tex]\( f(x) \)[/tex]:
[tex]\[ (x - (5 + i))(x - (5 - i)) \][/tex]
This product simplifies to:
[tex]\[ (x - 5 - i)(x - 5 + i) = (x - 5)^2 - i^2 = (x - 5)^2 + 1 \][/tex]
[tex]\[ = x^2 - 10x + 25 + 1 = x^2 - 10x + 26 \][/tex]
### Step 3: Perform Polynomial Division
We need to divide the polynomial [tex]\(f(x)\)[/tex] by the quadratic factor we just found:
[tex]\[ \frac{9x^3 - 97x^2 + 304x - 182}{x^2 - 10x + 26} \][/tex]
The quotient from this division is a linear polynomial (since a cubic divided by a quadratic leaves a linear), which we can form as:
[tex]\[ qx + r = 9x + p \][/tex]
In this case, we already know that the quotient is of the form [tex]\(9(x - 7/9)\)[/tex].
### Step 4: Find the Remaining Zero
Since:
[tex]\[ 9x^3 - 97x^2 + 304x - 182 = (x^2 - 10x + 26)(9x - 7) \][/tex]
Equating the remaining factor to zero gives us:
[tex]\[ 9x - 7 = 0 \Rightarrow x = \frac{7}{9} \][/tex]
### Step 5: List of All Zeros
The zeros of the polynomial are:
[tex]\[ 5 + i, 5 - i, \frac{7}{9} \][/tex]
### Step 6: Factor the Polynomial as Product of Linear Factors
Given the zeros, the polynomial can be factored as:
[tex]\[ f(x) = 9(x - (5+i))(x - (5-i))(x - \frac{7}{9}) \][/tex]
We can write the polynomial in its completely factored form:
[tex]\[ f(x) = 9 \left( x - 5 - i \right) \left( x - 5 + i \right) \left( x - \frac{7}{9} \right) \][/tex]
### Step 7: Simplify the Factors
[tex]\[ = 9[(x - 5)^2 + 1]\left( x - \frac{7}{9} \right) \][/tex]
### Step 8: Solve the Equation [tex]\( f(x) = 0 \)[/tex]
To solve [tex]\( f(x) = 0 \)[/tex], we set each factor to zero:
[tex]\[ (x - (5+i)) = 0 \Rightarrow x = 5 + i \][/tex]
[tex]\[ (x - (5-i)) = 0 \Rightarrow x = 5 - i \][/tex]
[tex]\[ (x - \frac{7}{9}) = 0 \Rightarrow x = \frac{7}{9} \][/tex]
### Summary
(a) All the zeros are: [tex]\( 5 + i \)[/tex], [tex]\( 5 - i \)[/tex], [tex]\(\frac{7}{9}\)[/tex].
(b) The polynomial factored as a product of linear factors is:
[tex]\[ f(x) = 9\left((x - (5 + i))(x - (5 - i))(x - \frac{7}{9})\right) \][/tex]
(c) The solutions to the equation [tex]\( f(x) = 0 \)[/tex] are:
[tex]\[ x = 5 + i, x = 5 - i, x = \frac{7}{9} \][/tex]
### Given:
[tex]\[ f(x) = 9x^3 - 97x^2 + 304x - 182 \][/tex]
A known zero is [tex]\(5 + i\)[/tex].
### Step 1: Identify the Conjugate Zero
Since the coefficients of the polynomial are real, the complex conjugate of any complex zero is also a zero. Therefore, if [tex]\(5 + i\)[/tex] is a zero, [tex]\(5 - i\)[/tex] must also be a zero.
### Step 2: Form the Quadratic Factor
The zeros [tex]\(5 + i\)[/tex] and [tex]\(5 - i\)[/tex] imply a quadratic factor of the polynomial [tex]\( f(x) \)[/tex]:
[tex]\[ (x - (5 + i))(x - (5 - i)) \][/tex]
This product simplifies to:
[tex]\[ (x - 5 - i)(x - 5 + i) = (x - 5)^2 - i^2 = (x - 5)^2 + 1 \][/tex]
[tex]\[ = x^2 - 10x + 25 + 1 = x^2 - 10x + 26 \][/tex]
### Step 3: Perform Polynomial Division
We need to divide the polynomial [tex]\(f(x)\)[/tex] by the quadratic factor we just found:
[tex]\[ \frac{9x^3 - 97x^2 + 304x - 182}{x^2 - 10x + 26} \][/tex]
The quotient from this division is a linear polynomial (since a cubic divided by a quadratic leaves a linear), which we can form as:
[tex]\[ qx + r = 9x + p \][/tex]
In this case, we already know that the quotient is of the form [tex]\(9(x - 7/9)\)[/tex].
### Step 4: Find the Remaining Zero
Since:
[tex]\[ 9x^3 - 97x^2 + 304x - 182 = (x^2 - 10x + 26)(9x - 7) \][/tex]
Equating the remaining factor to zero gives us:
[tex]\[ 9x - 7 = 0 \Rightarrow x = \frac{7}{9} \][/tex]
### Step 5: List of All Zeros
The zeros of the polynomial are:
[tex]\[ 5 + i, 5 - i, \frac{7}{9} \][/tex]
### Step 6: Factor the Polynomial as Product of Linear Factors
Given the zeros, the polynomial can be factored as:
[tex]\[ f(x) = 9(x - (5+i))(x - (5-i))(x - \frac{7}{9}) \][/tex]
We can write the polynomial in its completely factored form:
[tex]\[ f(x) = 9 \left( x - 5 - i \right) \left( x - 5 + i \right) \left( x - \frac{7}{9} \right) \][/tex]
### Step 7: Simplify the Factors
[tex]\[ = 9[(x - 5)^2 + 1]\left( x - \frac{7}{9} \right) \][/tex]
### Step 8: Solve the Equation [tex]\( f(x) = 0 \)[/tex]
To solve [tex]\( f(x) = 0 \)[/tex], we set each factor to zero:
[tex]\[ (x - (5+i)) = 0 \Rightarrow x = 5 + i \][/tex]
[tex]\[ (x - (5-i)) = 0 \Rightarrow x = 5 - i \][/tex]
[tex]\[ (x - \frac{7}{9}) = 0 \Rightarrow x = \frac{7}{9} \][/tex]
### Summary
(a) All the zeros are: [tex]\( 5 + i \)[/tex], [tex]\( 5 - i \)[/tex], [tex]\(\frac{7}{9}\)[/tex].
(b) The polynomial factored as a product of linear factors is:
[tex]\[ f(x) = 9\left((x - (5 + i))(x - (5 - i))(x - \frac{7}{9})\right) \][/tex]
(c) The solutions to the equation [tex]\( f(x) = 0 \)[/tex] are:
[tex]\[ x = 5 + i, x = 5 - i, x = \frac{7}{9} \][/tex]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.