Writing Polynomial Functions from Complex Roots

Question 1

Write an equation for the cubic polynomial function shown. To find the equation of the function, first find the rangedomain✔ zeroes of the graph.

Revolt · Accepted Answer

✔ zeroes

Question 2

Write an equation for the cubic polynomial function whose graph has zeroes at 2, 3, and 5. [image 1] Can any of the roots have multiplicity? [image 2] How can you find a function that has these roots?

Revolt · Accepted Answer

One such function is f(x) = (x − 2)(x − 3)(x − 5) = x³ − 10x² + 31x − 30. None of the roots can have multiplicity greater than 1 because the three distinct roots already account for the cubic’s degree. Write each root as a linear factor and multiply the factors.

Question 3

Write an equation for the cubic polynomial function whose graph has zeroes at 2, 3, and 5. Write the polynomial function for the graph.f(x) = (x – 2)(x – 3)(x – 5)Simplify the right side. What is the equation?

Revolt · Accepted Answer

f(x) = x3 – 10x2 + 31x – 30

Question 4

Suppose the graph of a cubic polynomial function has the same zeroes and passes through the coordinate (0, –5).Describe the steps for writing the equation of this cubic polynomial function.

Revolt · Accepted Answer

Use the zeroes to determine the roots. Write the polynomial as a product of the leading coefficient, a, and the factors, where each factor is x minus a root. Use the y-intercept (0, –5) to solve for the leading coefficient. Substitute the leading coefficient into the polynomial function for a and simplify.

Question 5

Which of the following did you include in your description?

Revolt · Accepted Answer

Use the zeroes to determine the roots.; Write the polynomial as a product of the leading coefficient, a, and the factors, where each factor is x minus a root.; Use the y-intercept (0, –5) to solve for the leading coefficient.; Substitute the leading coefficient into the polynomial function for a and simplify.

Question 6

Suppose the graph of a cubic polynomial function has the same zeroes and passes through the coordinate (0, –5).Write the equation of this cubic polynomial function. Recall that the zeroes are (2, 0), (3, 0), and (5, 0). What is the y-intercept of this graph?

Revolt · Accepted Answer

–5

Question 7

The linear factors of the cubic are ✔ (x – 2)(x – 3)(x – 5)(x – 2)(x – 3)(x – 5)(x + 5)(x + 2)(x + 3)(x + 5)(x + 2)(x + 3)(x – 5).

Revolt · Accepted Answer

✔ (x – 2)(x – 3)(x – 5)

Question 8

To solve for the leading coefficient, use a = –5(0 – 2)(0 – 3)(0 – 5) a = 5(0 – 2)(0 – 3)(0 – 5) 5 = a(0 – 2)(0 – 3)(0 – 5)✔ –5 = a(0 – 2)(0 – 3)(0 – 5).

Revolt · Accepted Answer

✔ –5 = a(0 – 2)(0 – 3)(0 – 5)

Question 9

The equation of the polynomial function is:

Revolt · Accepted Answer

Answer image: //media.edgenuity.com/EVresources/3111-06-07-25-05-task/mc002-4.jpg

Question 10

Now, suppose one of the roots of the polynomial function is irrational. The roots of the function are 2, , and 5. Write the equation for this polynomial function. Which of the following must also be a root of the function?

Revolt · Accepted Answer

Answer image: //media.edgenuity.com/EVresources/3111-06-07-25-06-task/mc001-2.jpg

Writing Polynomial Functions from Complex Roots Answers

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