Practice Test

Question 8 • Advanced Algebra: Concepts and Connections

A polynomial function has a root of negative 6 with multiplicity 3 and a root of 2 with multiplicity 4 If the function has a negative leading coefficient and is of odd degree, which could be the graph of the function?

Answer

Image with description On a coordinate plane with x and y axes marked from negative 8 to 8, a polynomial graph is drawn.It starts from quadrant 3 passing through (negative 8 comma negative 1 point 5), it touches x axis at (negative 6 comma 0) and moves down and curves up at (negative 0 point 5 comma negative 3). It then passes through x axis at (2 comma 0) and moves to quadrant 1 passing through (4 comma 8).

Image with description On a coordinate plane with x and y axes marked from negative 8 to 8, a polynomial graph is drawn. It starts from quadrant 2, passes through (negative 8 comma 8), crosses the x axis at (negative 6 comma 0), moves down and curves up at (negative 3 point 5 comma negative 3), and crosses the y axis at (0 comma negative 1). It then touches the x axis at (2 comma 0) and moves to quadrant 4 passing through (6 comma negative 8).

Image with description On a coordinate plane with x and y axes marked from negative 8 to 8, a polynomial graph is drawn. it starts from quadrant 2 passing through (negative 8 comma 0 point 9), touches the x axis at (negative 6 comma 0), it curves up and passes through y axis at (0 comma 1 point 5). It then crosses x axis at (2 comma 0) moves to quadrant 4 through (5 comma negative 8).

Image with description On a coordinate plane with x and y axes marked from negative 8 to 8, a polynomial graph is drawn. It starts from quadrant 1, passing through (5 comma 8), touches x axis at (2 comma 0), curves up and passes through y axis at (0 comma 2). It then moves up and curves down at (negative 3 point 5 comma 6), passes through x axis at (negative 6 comma 0) and moves to quadrant 3 passing through (negative 7 comma negative 8).

Explanation

The multiplicities add to 3 + 4 = 7, so the polynomial has odd degree. With a negative leading coefficient, its graph rises on the far left and falls on the far right. A root of multiplicity 3 at x = −6 has odd multiplicity, so the graph crosses the x-axis there; a root of multiplicity 4 at x = 2 has even multiplicity, so it touches the axis and turns around. Choice B has all of these features.

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