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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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Answer:

Q1 - D. f(x) = [tex]x^4-9x^2-50x-150[/tex]

Q13 - A. ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24.

Q14 - A. [tex]7x+6+4x^{2}[/tex]

Step-by-step explanation:

Question 1:

We know that rational roots always occurs in pairs. So, the zeros of the function will be 5, -3, -1+3i, -1-3i

So, the factored form is [tex](x-5)(x+3)(x+1-3i)(x+1+3i)=0[/tex]

i.e. [tex](x^2-2x-15)(x+1-3i)(x+1+3i)=0[/tex]

i.e. [tex](x^3-x^2-3ix^2-17x+6ix-15+45i)(x+1+3i)=0[/tex]

i.e. [tex]x^4-9x^2-50x-150=0[/tex]

Hence, the polynomial function is [tex]f(x)=x^4-9x^2-50x-150[/tex].

Question 13:  

Rational Zeros Theorem states that 'If p(x) is a polynomial with integer coefficients and if [tex]\frac{p}{q}[/tex] is a zero of p(x) = 0. Then, p is a factor of the constant term of p(x) and q is a factor of the leading coefficient of p(x)'.

Let, [tex]\frac{p}{q}[/tex] is a zero of [tex]x^3-7x^2+9x-24=0[/tex]. Then, p is a factor of -24 and q is a factor of 1.

Thus, possible values of p = ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24 and q = ±1

This gives, possible values of [tex]\frac{p}{q}[/tex] are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24.

Question 14:

We have, f(x) = 7x + 6 and g(x) = [tex]4x^{2}[/tex]

Then, (f+g)(x) = f(x) + g(x) =  7x + 6 + [tex]4x^{2}[/tex]

So, (f+g)(x) = [tex]7x+6+4x^{2}[/tex]

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