Question

A polynomial function has a root of zero with multiplicity one, and root of two with multiplicity four. If the function has a negative leading coefficient, and is of odd degree, which of the following is true.

- the function is positive on (-infinity, 0)

-the function is negative on (0,2)

-the function is negative on (2,infinity)

-the function is positive on (0, infinity)

Answer 1

A, B, and C

Explanation:

Had the same question on Edge :)

Answer 2

Answer: A, B, and C

Explanation:

root of zero (x = 0) with multiplicity of 1 (x = 0)¹ means that the graph CROSSES the axis at x = 0

root of two (x = 2) with multiplicity of 4 (x = 2)⁴ means that the graph TOUCHES the axis at x = 2

WHY?

• odd numbered multiplicity crosses the x-axis
• even numbered multiplicity touches the x-axis (it is a turning point)

Leading Coefficient (end behavior of right side):

• If the leading coefficient is positive, then right side goes to +∞
• If the leading coefficient is negative, then right side goes to -∞

Degree of polynomial (left side behavior):

• If degree is even, then left side is the same as the right side
• If the degree is odd, then left side is opposite of right side

The given polynomial has a negative leading coefficent and odd degree so the left side goes to +∞ and the right side goes to -∞

OBSERVE INTERVALS:

• (-∞, 0) is above the x-axis so POSITIVE
• (0, 2) is below the x-axis so NEGATIVE
• (2, +∞) is also below the x-axis x=2 was a turning point so NEGATIVE

See graph to confirm statements above