12)Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

5, -3, and -1 + 3i

f(x) = x4 + 12.5x2 - 50x - 150

f(x) = x4 - 4x3 + 15x2 + 25x + 150

f(x) = x4 - 4x3 - 15x2 - 25x - 150

f(x) = x4 - 9x2 - 50x - 150

Question 13(Multiple Choice Worth 5 points)

Use the Rational Zeros Theorem to write a list of all potential rational zeros.

f(x) = x3 - 7x2 + 9x - 24

±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24

±1, ±2, ±3, ±4, ±24

±1, ±one divided by two, ±2, ±3, ±4, ±6, ±8, ±12, ±24

±1, ±2, ±3, ±4, ±6, ±12, ±24

Question 14(Multiple Choice Worth 5 points)

Perform the requested operation or operations.

f(x) = 7x + 6, g(x) = 4x2

Find (f + g)(x).

7x + 6 + 4x2

28x3 + 24x

7x + 6 - 4x2

seven x plus six divided by four x squared.

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(12) Given zeroes are 5, -3 and -1 + 3i. So, the other conjugate of -1 + 3i, i.e., -1 - 3i will also be a root of the polynomial. So the polynomial f(x) will be of degree 4 and is given by

[tex]f(x)=(x-5)(x+3)(x+1-3i)(x+1+3i)\\\\\Rightarrow f(x)=(x^2-2x-15)(x^2+2x+1-9i^2)\\\\\Rightarrow f(x)=(x^2-2x-15)(x^2+2x+10)\\\\\Rightarrow f(x)=x^4-9x^2-50x-150.[/tex]

(13) Given polynomial is

[tex]f(x)=x^3-7x^2+9x-24.[/tex]

Now, we will substitute the rational numbers in place of 'x' and check whether the value of f(x) becomes zero or not.

We will see that

[tex]f(-1)\neq 0,~~f(1)\neq 0,~~f(2)\neq 0,~~f(-2)\neq 0, ~~f(3)\neq 0, ~~f(-3)\neq 0,~~etc[/tex]

Also, the polynomial is not zero for any rational number.

(14) Given, [tex]f(x)=7x+6~~\textup{and}~~g(x) = 4x^2.[/tex]

So,

[tex](f+g)(x)=f(x)+g(x)=7x+6+4x^2.[/tex]

Thus, the problems are solved.