Respuesta :
Given:
Polynomial of degree 3 that has zeros of 18, 10i, and −10i, and has a value of −34 when x=1.
To find:
The polynomial in standard form.
Solution:
If c is a zero of a polynomial then (x-a) is a factor of that polynomial.
18, 10i, and −10i are zeroes, so (x-18), (x-10i) and (x+10i) respectively are the factors of given polynomial.
So, the three degree polynomial is
[tex]P(x)=a(x-18)(x-10i)(x+10i)[/tex]
where, a is a constant.
[tex]P(x)=a(x-18)(x^2-(10i)^2)[/tex]
[tex]P(x)=a(x-18)(x^2-100i^2)[/tex]
[tex]P(x)=a(x-18)(x^2-100(-1))[/tex] [tex][\because i^2=-1][/tex]
[tex]P(x)=a(x-18)(x^2+100)[/tex]
[tex]P(x)=a(x^3-18x^2+100x-1800)[/tex] ...(i)
The value of P(x) is -34 at x=1.
[tex]-34=a((1)^3-18(1)^2+100(1)-1800)[/tex]
[tex]-34=a(1-18+100-1800)[/tex]
[tex]-34=1717a[/tex]
Divide both sides by 1717.
[tex]\dfrac{-34}{1717}=a[/tex]
[tex]\dfrac{-2}{101}=a[/tex]
Put this value in (i).
[tex]P(x)=\dfrac{-2}{101}(x^3-18x^2+100x-1800)[/tex]
[tex]P(x)=\dfrac{-2}{101}x^3+\dfrac{36}{101}x^2-\dfrac{200}{101}x+\dfrac{3600}{101}[/tex]
Therefore, the required polynomial is [tex]P(x)=\dfrac{-2}{101}x^3+\dfrac{36}{101}x^2-\dfrac{200}{101}x+\dfrac{3600}{101}[/tex].
The polynomial in standard form is: [tex]\mathbf{P(x) = \frac{2}{101}x^3 - \frac{36}{101}x^2 +\frac{200}{101}x - \frac{3600}{101}}[/tex]
The zeros of the polynomial are 18, 10i and -10i.
So, the polynomial is:
[tex]\mathbf{P(x) = a(x - 18)(x - 10i)(x -(-10i))}[/tex]
Open bracket
[tex]\mathbf{P(x) = a(x - 18)(x - 10i)(x +10i)}[/tex]
Apply difference of two squares
[tex]\mathbf{P(x) = a(x - 18)(x^2 - (10i)^2)}[/tex]
Expand
[tex]\mathbf{P(x) = a(x - 18)(x^2 - 100i^2)}[/tex]
In complex numbers, i^2 = -1.
So, we have:
[tex]\mathbf{P(x) = a(x - 18)(x^2 - 100(-1))}[/tex]
Expand
[tex]\mathbf{P(x) = a(x - 18)(x^2 + 100)}[/tex]
From the question, P(1) = -34.
So, we have:
[tex]\mathbf{a(1 - 18)(1^2 + 100) = -34}[/tex]
[tex]\mathbf{a(- 17)(1 + 100) = -34}[/tex]
Divide both sides by -17
[tex]\mathbf{a(1 + 100) = 2}[/tex]
[tex]\mathbf{a(101) = 2}[/tex]
Make (a) the subject
[tex]\mathbf{a = \frac{2}{101}}[/tex]
Substitute [tex]\mathbf{a = \frac{2}{101}}[/tex] in [tex]\mathbf{P(x) = a(x - 18)(x^2 + 100)}[/tex]
[tex]\mathbf{P(x) = \frac{2}{101}(x - 18)(x^2 + 100)}[/tex]
Expand
[tex]\mathbf{P(x) = \frac{2}{101}(x^3 - 18x^2 +100x - 1800)}[/tex]
Open bracket
[tex]\mathbf{P(x) = \frac{2}{101}x^3 - \frac{36}{101}x^2 +\frac{200}{101}x - \frac{3600}{101}}[/tex]
Hence, the polynomial in standard form is: [tex]\mathbf{P(x) = \frac{2}{101}x^3 - \frac{36}{101}x^2 +\frac{200}{101}x - \frac{3600}{101}}[/tex]
Read more about polynomials at:
https://brainly.com/question/11536910
