Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Let's construct the polynomial function \( f(x) \) given the specified roots and leading coefficient. We'll go through the process step-by-step:
1. Identify the roots and leading coefficient:
- Given roots: \( 5 \), \( \sqrt{5} \), and \( -\sqrt{5} \)
- Leading coefficient: \( 4 \)
2. Write the polynomial in factored form using the roots:
A root \( r \) of a polynomial can be represented by the factor \( (x - r) \). Therefore, the polynomial with the given roots can be expressed as:
[tex]\[ f(x) = 4(x - 5)(x - \sqrt{5})(x + \sqrt{5}) \][/tex]
3. Simplify the factored form:
Let's first deal with the two roots involving square roots. Notice:
[tex]\[ (x - \sqrt{5})(x + \sqrt{5}) = x^2 - (\sqrt{5})^2 = x^2 - 5 \][/tex]
Substituting back into the equation, we have:
[tex]\[ f(x) = 4(x - 5)(x^2 - 5) \][/tex]
4. Expand the expression:
We'll now expand \( (x - 5)(x^2 - 5) \):
[tex]\[ (x - 5)(x^2 - 5) = x(x^2 - 5) - 5(x^2 - 5) \][/tex]
[tex]\[ = x^3 - 5x - 5x^2 + 25 \][/tex]
Reordering terms for clarity:
[tex]\[ = x^3 - 5x^2 - 5x + 25 \][/tex]
5. Apply the leading coefficient:
Finally, multiply the expanded polynomial by the leading coefficient \( 4 \):
[tex]\[ f(x) = 4(x^3 - 5x^2 - 5x + 25) \][/tex]
[tex]\[ = 4x^3 - 20x^2 - 20x + 100 \][/tex]
So, the polynomial function \( f(x) \) in standard form is:
[tex]\[ f(x) = 4x^3 - 20x^2 - 20x + 100 \][/tex]
This is the polynomial function having a leading coefficient of 4 and the roots [tex]\( 5 \)[/tex], [tex]\( \sqrt{5} \)[/tex], and [tex]\( -\sqrt{5} \)[/tex].
1. Identify the roots and leading coefficient:
- Given roots: \( 5 \), \( \sqrt{5} \), and \( -\sqrt{5} \)
- Leading coefficient: \( 4 \)
2. Write the polynomial in factored form using the roots:
A root \( r \) of a polynomial can be represented by the factor \( (x - r) \). Therefore, the polynomial with the given roots can be expressed as:
[tex]\[ f(x) = 4(x - 5)(x - \sqrt{5})(x + \sqrt{5}) \][/tex]
3. Simplify the factored form:
Let's first deal with the two roots involving square roots. Notice:
[tex]\[ (x - \sqrt{5})(x + \sqrt{5}) = x^2 - (\sqrt{5})^2 = x^2 - 5 \][/tex]
Substituting back into the equation, we have:
[tex]\[ f(x) = 4(x - 5)(x^2 - 5) \][/tex]
4. Expand the expression:
We'll now expand \( (x - 5)(x^2 - 5) \):
[tex]\[ (x - 5)(x^2 - 5) = x(x^2 - 5) - 5(x^2 - 5) \][/tex]
[tex]\[ = x^3 - 5x - 5x^2 + 25 \][/tex]
Reordering terms for clarity:
[tex]\[ = x^3 - 5x^2 - 5x + 25 \][/tex]
5. Apply the leading coefficient:
Finally, multiply the expanded polynomial by the leading coefficient \( 4 \):
[tex]\[ f(x) = 4(x^3 - 5x^2 - 5x + 25) \][/tex]
[tex]\[ = 4x^3 - 20x^2 - 20x + 100 \][/tex]
So, the polynomial function \( f(x) \) in standard form is:
[tex]\[ f(x) = 4x^3 - 20x^2 - 20x + 100 \][/tex]
This is the polynomial function having a leading coefficient of 4 and the roots [tex]\( 5 \)[/tex], [tex]\( \sqrt{5} \)[/tex], and [tex]\( -\sqrt{5} \)[/tex].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.