Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To factor the polynomial [tex]\( f(x) = x^5 - 5x^4 - 13x^3 + 65x^2 + 36x - 180 \)[/tex] completely, we are given that [tex]\( x = 5 \)[/tex] is a root. Knowing that [tex]\( x = 5 \)[/tex] is a root, we can factor out [tex]\( (x - 5) \)[/tex] from [tex]\( f(x) \)[/tex]. Let's proceed with the step-by-step factorization:
### Step 1: Polynomial Division
Since [tex]\( x = 5 \)[/tex] is a root of [tex]\( f(x) \)[/tex], [tex]\( x - 5 \)[/tex] is a factor of [tex]\( f(x) \)[/tex]. We perform polynomial division of [tex]\( f(x) \)[/tex] by [tex]\( x - 5 \)[/tex].
1. Setup the problem:
[tex]\( f(x) = x^5 - 5x^4 - 13x^3 + 65x^2 + 36x - 180 \)[/tex]
We divide this by [tex]\( x - 5 \)[/tex].
2. Perform polynomial division:
By synthetic division or long division, we divide the polynomial by [tex]\( x - 5 \)[/tex]:
[tex]\[ \begin{array}{r|rrrrrr} 5 & 1 & -5 & -13 & 65 & 36 & -180 \\ \hline & 1 & 0 & -13 & 0 & 36 & 0 \\ \end{array} \][/tex]
We get the quotient as:
[tex]\[ x^4 - 13x^2 + 36 \][/tex]
So, we have:
[tex]\[ f(x) = (x - 5)(x^4 - 13x^2 + 36) \][/tex]
### Step 2: Factor the Quotient Polynomial
Next, we need to factor the quotient polynomial [tex]\( x^4 - 13x^2 + 36 \)[/tex].
We recognize it as a quadratic in terms of [tex]\( x^2 \)[/tex]. Let's rewrite [tex]\( x^4 - 13x^2 + 36 \)[/tex] as:
[tex]\[ y = x^2 \][/tex]
[tex]\[ y^2 - 13y + 36 \][/tex]
### Step 3: Solve the Quadratic [tex]\( y^2 - 13y + 36 \)[/tex]
To factor [tex]\( y^2 - 13y + 36 \)[/tex], we find the roots of the quadratic equation:
[tex]\[ y^2 - 13y + 36 = 0 \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{13 \pm \sqrt{13^2 - 4 \cdot 36}}{2} \][/tex]
[tex]\[ y = \frac{13 \pm \sqrt{169 - 144}}{2} \][/tex]
[tex]\[ y = \frac{13 \pm \sqrt{25}}{2} \][/tex]
[tex]\[ y = \frac{13 \pm 5}{2} \][/tex]
So, we have:
[tex]\[ y = 9 \quad \text{or} \quad y = 4 \][/tex]
In terms of [tex]\( x \)[/tex],
[tex]\[ x^2 = 9 \quad \text{or} \quad x^2 = 4 \][/tex]
[tex]\[ x = \pm 3 \quad \text{or} \quad x = \pm 2 \][/tex]
### Step 4: Write the Final Factorization
Therefore, we can factor [tex]\( x^4 - 13x^2 + 36 \)[/tex] as:
[tex]\[ x^4 - 13x^2 + 36 = (x^2 - 9)(x^2 - 4) = (x - 3)(x + 3)(x - 2)(x + 2) \][/tex]
### Step 5: Combine All Factors
Taking all the factors into account, the completely factored form of [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = (x - 5)(x - 3)(x - 2)(x + 2)(x + 3) \][/tex]
So, the polynomial [tex]\( f(x) \)[/tex] is factored completely as:
[tex]\[ f(x) = (x - 5)(x - 3)(x - 2)(x + 2)(x + 3) \][/tex]
### Step 1: Polynomial Division
Since [tex]\( x = 5 \)[/tex] is a root of [tex]\( f(x) \)[/tex], [tex]\( x - 5 \)[/tex] is a factor of [tex]\( f(x) \)[/tex]. We perform polynomial division of [tex]\( f(x) \)[/tex] by [tex]\( x - 5 \)[/tex].
1. Setup the problem:
[tex]\( f(x) = x^5 - 5x^4 - 13x^3 + 65x^2 + 36x - 180 \)[/tex]
We divide this by [tex]\( x - 5 \)[/tex].
2. Perform polynomial division:
By synthetic division or long division, we divide the polynomial by [tex]\( x - 5 \)[/tex]:
[tex]\[ \begin{array}{r|rrrrrr} 5 & 1 & -5 & -13 & 65 & 36 & -180 \\ \hline & 1 & 0 & -13 & 0 & 36 & 0 \\ \end{array} \][/tex]
We get the quotient as:
[tex]\[ x^4 - 13x^2 + 36 \][/tex]
So, we have:
[tex]\[ f(x) = (x - 5)(x^4 - 13x^2 + 36) \][/tex]
### Step 2: Factor the Quotient Polynomial
Next, we need to factor the quotient polynomial [tex]\( x^4 - 13x^2 + 36 \)[/tex].
We recognize it as a quadratic in terms of [tex]\( x^2 \)[/tex]. Let's rewrite [tex]\( x^4 - 13x^2 + 36 \)[/tex] as:
[tex]\[ y = x^2 \][/tex]
[tex]\[ y^2 - 13y + 36 \][/tex]
### Step 3: Solve the Quadratic [tex]\( y^2 - 13y + 36 \)[/tex]
To factor [tex]\( y^2 - 13y + 36 \)[/tex], we find the roots of the quadratic equation:
[tex]\[ y^2 - 13y + 36 = 0 \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{13 \pm \sqrt{13^2 - 4 \cdot 36}}{2} \][/tex]
[tex]\[ y = \frac{13 \pm \sqrt{169 - 144}}{2} \][/tex]
[tex]\[ y = \frac{13 \pm \sqrt{25}}{2} \][/tex]
[tex]\[ y = \frac{13 \pm 5}{2} \][/tex]
So, we have:
[tex]\[ y = 9 \quad \text{or} \quad y = 4 \][/tex]
In terms of [tex]\( x \)[/tex],
[tex]\[ x^2 = 9 \quad \text{or} \quad x^2 = 4 \][/tex]
[tex]\[ x = \pm 3 \quad \text{or} \quad x = \pm 2 \][/tex]
### Step 4: Write the Final Factorization
Therefore, we can factor [tex]\( x^4 - 13x^2 + 36 \)[/tex] as:
[tex]\[ x^4 - 13x^2 + 36 = (x^2 - 9)(x^2 - 4) = (x - 3)(x + 3)(x - 2)(x + 2) \][/tex]
### Step 5: Combine All Factors
Taking all the factors into account, the completely factored form of [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = (x - 5)(x - 3)(x - 2)(x + 2)(x + 3) \][/tex]
So, the polynomial [tex]\( f(x) \)[/tex] is factored completely as:
[tex]\[ f(x) = (x - 5)(x - 3)(x - 2)(x + 2)(x + 3) \][/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
