Question

writing polynomial functions from complex roots which polynomial function has a leading coefficient of 1 and roots (7 + i) and (5 − i) with multiplicity 1? \\( f(x) = (x + (7 - i))(x + (5 + i))(x + (7 + i))(x + (5 - i)) \\) \\( f(x) = (x + 7)(x - i)(x - 5)(x + i) \\) \\( f(x) = (x + 7)(x - i)(x + 5)(x + i) \\) \\( f(x) = (x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i)) \\)

Explanation:

Step1: Recall Complex Conjugate Root Theorem

For a polynomial with real coefficients, if \(a + bi\) is a root, then \(a - bi\) is also a root. Given roots \((7 + i)\) and \((5 - i)\), their conjugates \((7 - i)\) and \((5 + i)\) must also be roots (since leading coefficient is real, 1 here).

Step2: Form the Polynomial from Roots

If \(r\) is a root of a polynomial \(f(x)\), then \((x - r)\) is a factor. So the factors corresponding to roots \(7 + i\), \(7 - i\), \(5 - i\), \(5 + i\) are \((x - (7 + i))\), \((x - (7 - i))\), \((x - (5 - i))\), \((x - (5 + i))\).

Step3: Identify the Correct Polynomial

The polynomial with these factors (and leading coefficient 1) is \(f(x)=(x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i))\), which matches the last option (the fourth one, let's check the options: the last option shown is \(f(x)=(x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i))\) when we parse the text, wait, looking at the options:

Wait the options:

First option (leftmost): \(f(x)=(x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i))\) – wait no, let's re-express:

Wait the roots are \(7 + i\), \(7 - i\), \(5 - i\), \(5 + i\). So factors are \((x - (7 + i))\), \((x - (7 - i))\), \((x - (5 - i))\), \((x - (5 + i))\). So the polynomial is \((x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i))\), which is the leftmost option? Wait no, let's check the given options:

Looking at the image:

First (leftmost) option: \(f(x)=(x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i))\)

Second option: \(f(x)=(x + (7 - i))(x + (5 + i))(x + (7 + i))(x + (5 - i))\) – no, that's wrong (signs).

Third option: \(f(x)=(x + 7)(x - i)(x - 5)(x + i)\) – wrong roots.

Fourth option (rightmost): Wait no, the leftmost is the fourth? Wait the options are:

1. \(f(x)=(x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i))\)

1. \(f(x)=(x - 7)(x - i)(x - 5)(x + i)\)

1. \(f(x)=(x + 7)(x - i)(x + 5)(x + i)\)

1. \(f(x)=(x + (7 - i))(x + (5 + i))(x + (7 + i))(x + (5 - i))\) – no, that's adding.

Wait no, I think I mixed up. Wait the correct polynomial is formed by subtracting each root. So for root \(7 + i\), factor is \(x - (7 + i)\); for \(7 - i\), \(x - (7 - i)\); for \(5 - i\), \(x - (5 - i)\); for \(5 + i\), \(x - (5 + i)\). So the product is \((x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i))\), which is the leftmost option (the first one in the vertical list, the leftmost box). Wait the user's image shows:

Leftmost option: \(f(x)=(x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i))\)

Then next: \(f(x)=(x - 7)(x - i)(x - 5)(x + i)\)

Then: \(f(x)=(x + 7)(x - i)(x + 5)(x + i)\)

Rightmost: \(f(x)=(x + (7 - i))(x + (5 + i))(x + (7 + i))(x + (5 - i))\)

So the correct one is the leftmost option, because it has the factors with subtracting the roots (including conjugates), leading to the polynomial with roots \(7 + i\), \(7 - i\), \(5 - i\), \(5 + i\) and leading coefficient 1.

Answer:

The polynomial function is \(f(x)=(x - (7 - i))(x - (5 + i))(x - (7 + i))(x - (5 - i))\) (the left - most option among the given ones, which is the first vertical boxed option in the image).