Theorems About Roots of Polynomial Equations

List all the possible rational zeros
h(x) = x³ - 5x²+2x+12

• 1,2,3,4,6,12

• 1,2,6,12

• 1,2,3,4,6,12,-1,-2,-3,-4,-6,-12

A polynomial function with rational coefficients has the follow zeros: -1 + i and √5 . Find all additional zeros.

• 1 + i

• -√5 and 1 - i

• -√5 and -1 - i

• -1 - i

A polynomial function with rational coefficients has the follow zeros: −2 + √10  and -2. Find all additional zeros.

• 2 + √10

• 2

• 2 and -2 - √10

• -2 - √10

How many positive solutions does the following polynomial have? f(x) =3x⁵ - x⁴ + 3x² - 6x +11

• 4, 2, or 0

• 4

• 3 or 1

• 3

Use Descartes Rule of Signs to determine the possible number of positive and negative roots.

• positive, 2 or 0 negative

• positive, 1 negative

• 0 positive, 1 negative

• What?

Use Descartes Rule of Signs to determine the possible number of positive and negative roots.

• 3 or 1 positive roots and 0 negative roots

• 4, 2, or 0 positive roots and 1 negative root

• 4, 2, or 0 positive roots and 0 negative roots

• What?

Use the remainder theorem to evaluate the polynomial when x=3

• -405

• 405

• 396

• 395

Is (x-2) a factor of             f(x)= x³-8x²+14x-4?

• Yes, (x-2) is a factor. There is a remainder.

• No, (x-2) is  not a factor. The remainder is zero.

• Yes, (x-2) is a factor. The remainder is zero.

• No, (x-2) is  not a factor. There is a remainder. 

What are the zeros (x-intercepts)?

• x = 4, 7

• x = -1, 4/7

• x = -4/7, 1

• x = 7, -4

Use the Complex Conjugate Root Theorem to state another complex solution. Given that -2-5i is a root. 

• -2-5i

• -2+5i

• 2+5i

• 5i